{"id":108,"date":"2025-04-09T17:09:41","date_gmt":"2025-04-09T17:09:41","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-2-linear-equations-in-one-variable-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:29","modified_gmt":"2026-03-19T15:56:29","slug":"2-2-linear-equations-in-one-variable","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-2-linear-equations-in-one-variable\/","title":{"raw":"2.2 Linear Equations in One Variable","rendered":"2.2 Linear Equations in One Variable"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_eb6fde5e-763a-412f-8aa4-866649f706d5\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"ui-has-child-title\" data-type=\"abstract\"><header>\r\n<h2 data-type=\"title\">Learning Objectives<\/h2>\r\n<\/header><section>\r\n<p id=\"para-00001\">In this section, you will:<\/p>\r\n\r\n<ul id=\"list-00001\">\r\n \t<li>Solve equations in one variable algebraically.<\/li>\r\n \t<li>Solve a rational equation.<\/li>\r\n \t<li>Find a linear equation.<\/li>\r\n \t<li>Given the equations of two lines, determine whether their graphs are parallel or perpendicular.<\/li>\r\n \t<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<p id=\"fs-id1402487\">Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00\/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_001\">Figure 1<\/a>.<\/p>\r\n\r\n<div id=\"Figure_02_02_001\" class=\"os-figure\">\r\n<figure class=\"medium\" data-id=\"Figure_02_02_001\"><span id=\"fs-id1294390\" data-type=\"media\" data-alt=\"Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).\" width=\"509\" height=\"352\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/5e03adc287df01ae9cc9debe0416a2a49351b431\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<section id=\"fs-id2454936\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Linear Equations in One Variable<\/h2>\r\n<p id=\"fs-id1325030\">A <span id=\"term-00001\" data-type=\"term\">linear equation<\/span> is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and are solved using basic algebraic operations.\r\n<p id=\"fs-id2440186\">We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An <span id=\"term-00002\" data-type=\"term\">identity equation<\/span> is true for all values of the variable. Here is an example of an identity equation.<\/p>\r\n\r\n<div id=\"fs-id1384930\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1472975\">The <span id=\"term-00003\" data-type=\"term\">solution set<\/span> consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>will make the equation true.\r\n<p id=\"fs-id2906254\">A <span id=\"term-00004\" data-type=\"term\">conditional equation<\/span> is true for only some values of the variable. For example, if we are to solve the equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we have the following:\r\n<div id=\"fs-id1332285\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1518040\">The solution set consists of one number:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>It is the only solution and, therefore, we have solved a conditional equation.\r\n<p id=\"fs-id1476044\">An <span id=\"term-00005\" data-type=\"term\">inconsistent equation<\/span> results in a false statement. For example, if we are to solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>5<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>5<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we have the following:\r\n<div id=\"fs-id722242\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><mn>5<\/mn><mi>x<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>\u2260<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221220<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>False\u00a0statement<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><mn>5<\/mn><mi>x<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>\u2260<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221220<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>False\u00a0statement<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1554254\">Indeed,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u221215<\/mn><mo>\u2260<\/mo><mspace width=\"0.5em\"><\/mspace><mn>\u221220.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u221215<\/mn><mo>\u2260<\/mo><mspace width=\"0.5em\"><\/mspace><mn>\u221220.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There is no solution because this is an inconsistent equation.\r\n<p id=\"fs-id2279267\">Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.<\/p>\r\n\r\n<div id=\"fs-id1549902\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Linear Equation in One Variable<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1943551\">A linear equation in one variable can be written in the form<\/p>\r\n\r\n<div id=\"fs-id2519610\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1511576\">where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b <\/em>are real numbers,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1592386\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1337817\"><strong>Given a linear equation in one variable, use algebra to solve it.<\/strong><\/p>\r\n<p id=\"fs-id1759614\">The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mo>_________,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mo>_________,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>if <em data-effect=\"italics\">x <\/em>is the unknown. There is no set order, as the steps used depend on what is given:\r\n<ol id=\"fs-id1334010\" type=\"1\">\r\n \t<li>We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.<\/li>\r\n \t<li>Apply the distributive property as needed:\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Isolate the variable on one side of the equation.<\/li>\r\n \t<li>When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_01\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id3264315\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1294630\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Equation in One Variable<\/h3>\r\n<p id=\"fs-id1778894\">Solve the following equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>19.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>19.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id780975\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1215275\">This equation can be written in the form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>by subtracting\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>19<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>19<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>from both sides. However, we may proceed to solve the equation in its original form by performing algebraic operations.\r\n<div id=\"fs-id1429109\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>19<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a07\u00a0from\u00a0both\u00a0sides<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0divide\u00a0by\u00a02<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>19<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a07\u00a0from\u00a0both\u00a0sides<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0divide\u00a0by\u00a02<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1793429\">The solution is 6.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1531582\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2980056\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1954826\">Solve the linear equation in one variable:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>\u22129.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>\u22129.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_02\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1542084\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2439455\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Equation Algebraically When the Variable Appears on Both Sides<\/h3>\r\n<p id=\"fs-id2726928\">Solve the following equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mn>12<\/mn><mo>=<\/mo><mn>15<\/mn><mn>\u22125<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mn>12<\/mn><mo>=<\/mo><mn>15<\/mn><mn>\u22125<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2439343\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1943089\">Apply standard algebraic properties.<\/p>\r\n\r\n<div id=\"fs-id2443828\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Apply\u00a0the\u00a0distributive\u00a0property<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>9<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Place\u00a0<\/mtext><mi>x<\/mi><mtext>-terms\u00a0on\u00a0one\u00a0side\u00a0and\u00a0simplify<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mrow><mn>15<\/mn><\/mrow><mn>9<\/mn><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac>\r\n<mn>1<\/mn>\r\n<mn>9<\/mn>\r\n<\/mfrac>\r\n<mtext>,\u00a0the\u00a0reciprocal\u00a0of\u00a09<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Apply\u00a0the\u00a0distributive\u00a0property<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>9<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Place\u00a0<\/mtext><mi>x<\/mi><mtext>-terms\u00a0on\u00a0one\u00a0side\u00a0and\u00a0simplify<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mrow><mn>15<\/mn><\/mrow><mn>9<\/mn><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac>\r\n<mn>1<\/mn>\r\n<mn>9<\/mn>\r\n<\/mfrac>\r\n<mtext>,\u00a0the\u00a0reciprocal\u00a0of\u00a09<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1333518\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\r\n<p id=\"fs-id1469028\">This problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach the final line,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1533557\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2439622\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1287709\">Solve the equation in one variable:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1352666\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving a Rational Equation<\/h2>\r\n<p id=\"fs-id2697408\">In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a <strong>rational equation<\/strong>.<\/p>\r\n<p id=\"fs-id1568247\">Recall that a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">rational number<\/span> is the ratio of two numbers, such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>A <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">rational expression<\/span> is the ratio, or quotient, of two polynomials. Here are three examples.\r\n<div id=\"fs-id2503300\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2507716\">Rational equations have a variable in the denominator in at least one of the terms.\r\nOur goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">least common denominator<\/span> (LCD).<\/p>\r\n<p id=\"fs-id1290385\">Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.<\/p>\r\n\r\n<div id=\"Example_02_02_03\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2707414\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1275451\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Rational Equation<\/h3>\r\n<p id=\"fs-id2500534\">Solve the rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>22<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>22<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2364839\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2708670\">We have three denominators;<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and 3. The LCD must contain\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and 3. An LCD of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>contains all three denominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the equation by the LCD\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1149071\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Use\u00a0the\u00a0distributive\u00a0property<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>3<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mn>6<\/mn>\r\n<\/menclose>\r\n<mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Cancel\u00a0out\u00a0the\u00a0common\u00a0factors<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0remaining\u00a0factors\u00a0by\u00a0each\u00a0numerator<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>21<\/mn><mo>\u2212<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>11<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mrow><mn>11<\/mn><\/mrow><mrow><mn>44<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Use\u00a0the\u00a0distributive\u00a0property<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>3<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mn>6<\/mn>\r\n<\/menclose>\r\n<mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Cancel\u00a0out\u00a0the\u00a0common\u00a0factors<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0remaining\u00a0factors\u00a0by\u00a0each\u00a0numerator<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>21<\/mn><mo>\u2212<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>11<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mrow><mn>11<\/mn><\/mrow><mrow><mn>44<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2382754\">A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial\u2014two terms added or subtracted\u2014such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Always consider a binomial as an individual factor\u2014the terms cannot be separated. For example, suppose a problem has three terms and the denominators are\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>First, factor all denominators. We then have\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>The\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>in the first denominator is separate from the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>in the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>one factor of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the 3. Thus, the LCD is the following:\r\n<div id=\"fs-id2268029\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1510962\">So, both sides of the equation would be multiplied by<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.\r\n<p id=\"fs-id1780189\">Another example is a problem with two denominators, such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Once the second denominator is factored as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>there is a common factor of <em data-effect=\"italics\">x<\/em> in both denominators and the LCD is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1829569\">Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.<\/p>\r\n\r\n<div id=\"fs-id1467328\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mi>a<\/mi>\r\n<mi>b<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>d<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mi>a<\/mi>\r\n<mi>b<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>d<\/mi>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id914245\">We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.<\/p>\r\n<span id=\"fs-id2119795\" data-type=\"media\" data-alt=\"\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"\" width=\"487\" height=\"43\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7d9d5ac2f0af14cb4dab1f5490e09fe724be0e36\" \/>\r\n<\/span>\r\n<p id=\"fs-id2639399\">Multiply<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mrow><mo>(<\/mo>\r\n<mi>d<\/mi>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mrow><mo>(<\/mo>\r\n<mi>d<\/mi>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mrow><mo>(<\/mo>\r\n<mi>c<\/mi>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mrow><mo>(<\/mo>\r\n<mi>c<\/mi>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>which results in\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>d<\/mi><mo>=<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>d<\/mi><mo>=<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1551812\">Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.<\/p>\r\n\r\n<div id=\"fs-id1931811\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Rational Equations<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2629095\">A <span id=\"term-00009\" data-type=\"term\">rational equation<\/span> contains at least one rational expression where the variable appears in at least one of the denominators.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1147894\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1519809\"><strong>Given a rational equation, solve it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2667782\" type=\"1\">\r\n \t<li>Factor all denominators in the equation.<\/li>\r\n \t<li>Find and exclude values that set each denominator equal to zero.<\/li>\r\n \t<li>Find the LCD.<\/li>\r\n \t<li>Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left.<\/li>\r\n \t<li>Solve the remaining equation.<\/li>\r\n \t<li>Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_04\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1831988\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1504300\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Rational Equation without Factoring<\/h3>\r\n<p id=\"fs-id2435055\">Solve the following rational equation:<\/p>\r\n\r\n<div id=\"fs-id863002\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2502004\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id3141796\">We have three denominators:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>No factoring is required. The product of the first two denominators is equal to the third denominator, so, the LCD is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Only one value is excluded from a solution set, 0.\r\n\r\n<math display=\"inline\"><semantics><mrow>&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>Next, multiply the whole equation (both sides of the equal sign) by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2503271\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd><mrow><mrow><mo>(<\/mo> <mrow><mfrac><mn>7<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mi>x<\/mi>\r\n<\/menclose>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mi>x<\/mi>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0<\/mtext><mn>2<\/mn><mi>x<\/mi><mtext>.<\/mtext><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Denominators\u00a0cancel\u00a0out<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22123<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>3<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mtext>or<\/mtext><mspace width=\"0.3em\"><\/mspace><mrow><mo>{<\/mo> <mrow>\r\n<mn>\u22121<\/mn><\/mrow> <mo>}<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd><mrow><mrow><mo>(<\/mo> <mrow><mfrac><mn>7<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mi>x<\/mi>\r\n<\/menclose>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mi>x<\/mi>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0<\/mtext><mn>2<\/mn><mi>x<\/mi><mtext>.<\/mtext><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Denominators\u00a0cancel\u00a0out<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22123<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>3<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mtext>or<\/mtext><mspace width=\"0.3em\"><\/mspace><mrow><mo>{<\/mo> <mrow>\r\n<mn>\u22121<\/mn><\/mrow> <mo>}<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1496122\">The proposed solution is \u22121,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>which is not an excluded value, so the solution set contains one number,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn> \u22121<\/mn><mo>, <\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn> \u22121<\/mn><mo>, <\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>written in set notation.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1254485\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1476079\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1476080\">Solve the rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_05\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1533182\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1595938\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Rational Equation by Factoring the Denominator<\/h3>\r\n<p id=\"fs-id3268868\">Solve the following rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1204080\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2436750\">First find the common denominator. The three denominators in factored form are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>,<\/mo><mn>10<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>,<\/mo><mn>10<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The smallest expression that is divisible by each one of the denominators is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Only\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>is an excluded value. Multiply the whole equation by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1581638\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>20<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>20<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>35<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>35<\/mn><\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>20<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>20<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>20<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>35<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>35<\/mn><\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2437669\">The solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1844482\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1768288\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1768289\">Solve the rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_06\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1536446\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2682405\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving Rational Equations with a Binomial in the Denominator<\/h3>\r\n<p id=\"fs-id1723264\">Solve the following rational equations and state the excluded values:<\/p>\r\n\r\n<ol id=\"fs-id1500286\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1216224\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<ol id=\"fs-id1996554\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<p id=\"fs-id833086\">The denominators<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>have nothing in common. Therefore, the LCD is the product\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>However, for this problem, we can cross-multiply.\r\n<div id=\"fs-id1447286\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mn>3<\/mn><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mfrac><mn>5<\/mn><mi>x<\/mi><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Distribute<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22122<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221230<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mn>3<\/mn><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mfrac><mn>5<\/mn><mi>x<\/mi><\/mfrac><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Distribute<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22122<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221230<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2493985\">The solution is 15.<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>The excluded values are\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<p id=\"fs-id2434774\">The LCD is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply both sides of the equation by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1901302\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/menclose>\r\n<mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/menclose>\r\n<mn>5<\/mn><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/menclose>\r\n<mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/menclose>\r\n<mn>5<\/mn><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<menclose notation=\"updiagonalstrike\">\r\n<mn>2<\/mn>\r\n<\/menclose>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1783284\">The solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>13<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>13<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The excluded value is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<p id=\"fs-id3156711\">The least common denominator is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply both sides of the equation by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1289204\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1923900\">The solution is 4. The excluded value is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1686117\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2869362\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2869363\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State the excluded values.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_07\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1823223\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1387285\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Rational Equation with Factored Denominators and Stating Excluded Values<\/h3>\r\n<p id=\"fs-id3094632\">Solve the rational equation after factoring the denominators:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State the excluded values.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1527395\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2437930\">We must factor the denominator<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22121.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22121.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We recognize this as the difference of squares, and factor it as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Thus, the LCD that contains each denominator is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.\r\n<div id=\"fs-id1958485\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0negative\u00a0sign<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0negative\u00a0sign<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1847437\">The solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>The excluded values are\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>1<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>1<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1475570\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2507037\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1425920\">Solve the rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2673660\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding a Linear Equation<\/h2>\r\n<p id=\"fs-id2803187\">Perhaps the most familiar form of a linear equation is the slope-intercept form, written as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mtext>slope<\/mtext> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mtext>slope<\/mtext> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mi>y<\/mi><mtext>-intercept<\/mtext><mtext>.<\/mtext> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mi>y<\/mi><mtext>-intercept<\/mtext><mtext>.<\/mtext> <\/mrow><\/annotation-xml><\/semantics><\/math>Let us begin with the slope.\r\n\r\n<section id=\"fs-id2737079\" data-depth=\"2\">\r\n<h3 data-type=\"title\">The Slope of a Line<\/h3>\r\n<p id=\"fs-id1539119\">The <span id=\"term-00010\" data-type=\"term\">slope<\/span> of a line refers to the ratio of the vertical change in <em data-effect=\"italics\">y<\/em> over the horizontal change in <em data-effect=\"italics\">x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\r\n\r\n<div id=\"fs-id1920678\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1525950\">If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_002\">Figure 2<\/a>. The lines indicate the following slopes:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"Figure_02_02_002\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_002\"><span id=\"fs-id1939377\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/143ef80bb95f29cd295ad08addad53ad6864937a\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">2<\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1467947\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Slope of a Line<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2512513\">The slope of a line, <em data-effect=\"italics\">m<\/em>, represents the change in <em data-effect=\"italics\">y<\/em> over the change in <em data-effect=\"italics\">x.<\/em> Given two points,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>the following formula determines the slope of a line containing these points:\r\n<div id=\"fs-id1908930\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_08\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2388781\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2388784\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Slope of a Line Given Two Points<\/h3>\r\n<p id=\"fs-id2952969\">Find the slope of a line that passes through the points<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2715020\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2715022\">We substitute the <em data-effect=\"italics\">y-<\/em>values and the <em data-effect=\"italics\">x-<\/em>values into the formula.<\/p>\r\n\r\n<div id=\"fs-id3234414\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mn>\u22127<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>7<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mn>\u22127<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>7<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1568234\">The slope is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>7<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>7<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1560357\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\r\n<p id=\"fs-id1477668\">It does not matter which point is called<\/p>\r\n<p id=\"fs-id1477666\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub><\/mrow><\/mrow><\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n<mo>)<\/mo>\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub><\/mrow><\/mrow><\/mrow><\/annotation-xml>\r\n\r\n<mo>)<\/mo>\r\n\r\nor\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub><\/mrow><\/mrow><\/mrow><\/mrow><\/semantics><\/math><mo>)<\/mo><mo>.<\/mo>\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub><\/mrow><\/mrow><\/mrow><\/annotation-xml>\r\n\r\n<mo>)<\/mo><mo>.<\/mo>\r\n\r\nAs long as we are consistent with the order of the <em data-effect=\"italics\">y<\/em> terms and the order of the <em data-effect=\"italics\">x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1225501\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1553581\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1482316\">Find the slope of the line that passes through the points<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_09\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1424004\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1581998\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Identifying the Slope and <em data-effect=\"italics\">y-<\/em>intercept of a Line Given an Equation<\/h3>\r\n<p id=\"fs-id1386856\">Identify the slope and <em data-effect=\"italics\">y-<\/em>intercept, given the equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2517224\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1763671\">As the line is in<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>form, the given line has a slope of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The <em data-effect=\"italics\">y-<\/em>intercept is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>\u22124.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>\u22124.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1918796\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\r\n<p id=\"eip-id1266666\">The <em data-effect=\"italics\">y<\/em>-intercept is the point at which the line crosses the <em data-effect=\"italics\">y-<\/em>axis. On the <em data-effect=\"italics\">y-<\/em>axis,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We can always identify the <em data-effect=\"italics\">y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em data-effect=\"italics\">b.<\/em> Or, just substitute\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and solve for <em data-effect=\"italics\">y.<\/em>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1520430\" data-depth=\"2\">\r\n<h3 data-type=\"title\">The Point-Slope Formula<\/h3>\r\n<p id=\"fs-id1560189\">Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.<\/p>\r\n\r\n<div id=\"fs-id1539802\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2497497\">This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.<\/p>\r\n\r\n<div id=\"fs-id1401850\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Point-Slope Formula<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1926574\">Given one point and the slope, the point-slope formula will lead to the equation of a line:<\/p>\r\n\r\n<div id=\"fs-id1939371\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_09a\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1780029\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1780031\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Equation of a Line Given the Slope and One Point<\/h3>\r\n<p id=\"fs-id1517411\">Write the equation of the line with slope<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the final equation in slope-intercept form.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1557465\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1562535\">Using the point-slope formula, substitute<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>for <em data-effect=\"italics\">m <\/em>and the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1537532\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>20<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>20<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id2736677\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\r\n<p id=\"fs-id2410998\">Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1939337\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1814119\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1814120\">Given<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>find the equation of the line in slope-intercept form passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_10\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">11<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2279286\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2694098\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Equation of a Line Passing Through Two Given Points<\/h3>\r\n<p id=\"fs-id1517573\">Find the equation of the line passing through the points<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the final equation in slope-intercept form.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1920249\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1920251\">First, we calculate the slope using the slope formula and two points.<\/p>\r\n\r\n<div id=\"fs-id1786245\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1539521\">Next, we use the point-slope formula with the slope of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and either point. Let\u2019s pick the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1572614\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0<\/mtext><mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0<\/mtext><mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1914286\">In slope-intercept form, the equation is written as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1592294\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\r\n<p id=\"fs-id1940795\">To prove that either point can be used, let us use the second point<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and see if we get the same equation.\r\n<div id=\"fs-id1790167\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1562421\">We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2520904\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Standard Form of a Line<\/h3>\r\n<p id=\"fs-id2508791\">Another way that we can represent the equation of a line is in <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">standard form<\/span>. Standard form is given as<\/p>\r\n\r\n<div id=\"fs-id3150952\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1422421\">where<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>A<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>A<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>C<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>C<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>are integers. The <em data-effect=\"italics\">x- <\/em>and <em data-effect=\"italics\">y-<\/em>terms are on one side of the equal sign and the constant term is on the other side.\r\n<div id=\"Example_02_02_11\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">12<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2431257\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2431259\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Equation of a Line and Writing It in Standard Form<\/h3>\r\n<p id=\"fs-id2736527\">Find the equation of the line with<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22126<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>\u22126<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the equation in standard form.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1388008\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2753839\">We begin using the point-slope formula.<\/p>\r\n\r\n<div id=\"fs-id1353048\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22126<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22126<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1357955\">From here, we multiply through by 2, as no fractions are permitted in standard form, and then move both variables to the left aside of the equal sign and move the constants to the right.<\/p>\r\n\r\n<div id=\"fs-id1315939\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1385368\">This equation is now written in standard form.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2426862\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2820293\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2820294\">Find the equation of the line in standard form with slope<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1548365\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Vertical and Horizontal Lines<\/h3>\r\n<p id=\"fs-id3207599\">The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">vertical line<\/span> is given as<\/p>\r\n\r\n<div id=\"fs-id2002088\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2386398\">where <em data-effect=\"italics\">c <\/em>is a constant. The slope of a vertical line is undefined, and regardless of the <em data-effect=\"italics\">y-<\/em>value of any point on the line, the <em data-effect=\"italics\">x-<\/em>coordinate of the point will be <em data-effect=\"italics\">c<\/em>.<\/p>\r\n<p id=\"fs-id1569566\">Suppose that we want to find the equation of a line containing the following points:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>\u22125<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>\u22125<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>First, we will find the slope.\r\n<div id=\"fs-id1400578\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>0<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>0<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1709161\">Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the <em data-effect=\"italics\">x-<\/em>coordinates are the same and we find a vertical line through<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>\u22123.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>\u22123.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_003\">Figure 3<\/a><strong>.<\/strong>\r\n<p id=\"fs-id2503257\">The equation of a <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">horizontal line<\/span> is given as<\/p>\r\n\r\n<div id=\"fs-id1477479\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1513867\">where <em data-effect=\"italics\">c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em data-effect=\"italics\">x-<\/em>value of a point on the line, the <em data-effect=\"italics\">y-<\/em>coordinate will be <em data-effect=\"italics\">c<\/em>.<\/p>\r\n<p id=\"fs-id2528940\">Suppose we want to find the equation of a line that contains the following set of points:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>We can use the point-slope formula. First, we find the slope using any two points on the line.\r\n<div id=\"fs-id1257613\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>0<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>0<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1258397\">Use any point for<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msub>\r\n<mi>x<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>,<\/mo><msub>\r\n<mi>y<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>in the formula, or use the <em data-effect=\"italics\">y<\/em>-intercept.\r\n<div id=\"fs-id1691149\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>0<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>0<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id3257368\">The graph is a horizontal line through<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>\u22122.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>\u22122.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>Notice that all of the <em data-effect=\"italics\">y-<\/em>coordinates are the same. See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_003\">Figure 3<\/a>.\r\n<div id=\"Figure_02_02_003\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_003\"><span id=\"fs-id1336884\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/2d21dfa5c23edca204827fda8f8f28b5312737a5\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">3<\/span>\r\n<span class=\"os-caption\">The line <em data-effect=\"italics\">x<\/em> = \u22123 is a vertical line. The line <em data-effect=\"italics\">y<\/em> = \u22122 is a horizontal line.<\/span><\/div>\r\n<\/div>\r\n<div id=\"Example_02_02_12\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">13<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2797158\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2797160\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Equation of a Line Passing Through the Given Points<\/h3>\r\n<p id=\"fs-id1273666\">Find the equation of the line passing through the given points:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1836962\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2764547\">The <em data-effect=\"italics\">x-<\/em>coordinate of both points is 1. Therefore, we have a vertical line,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1813611\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1538834\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1538835\">Find the equation of the line passing through<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><section id=\"fs-id1495944\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Whether Graphs of Lines are Parallel or Perpendicular<\/h2>\r\n<p id=\"fs-id1973968\">Parallel lines have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts. Lines that are <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">parallel<\/span> to each other will never intersect. For example, <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_004\">Figure 4<\/a> shows the graphs of various lines with the same slope,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>2.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mn>2.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"Figure_02_02_004\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_004\"><span id=\"fs-id1847446\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\" width=\"487\" height=\"593\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/346b578e36322d966a04b3ec8131b4f43419e060\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">4<\/span>\r\n<span class=\"os-caption\"> Parallel lines<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1929226\">All of the lines shown in the graph are parallel because they have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts.<\/p>\r\n<p id=\"fs-id1955734\">Lines that are <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">perpendicular<\/span> intersect to form a<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>90\u00b0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>90\u00b0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>-angle. The slope of one line is the negative <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">reciprocal<\/span> of the other. We can show that two lines are perpendicular if the product of the two slopes is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>:<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>\u22c5<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22121<\/mn><mo>:<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>\u22c5<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>For example, <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_005\">Figure 5<\/a> shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2527184\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><msub><mi>m<\/mi><mn>1<\/mn><\/msub><mo>\u22c5<\/mo><msub><mi>m<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><msub><mi>m<\/mi><mn>1<\/mn><\/msub><mo>\u22c5<\/mo><msub><mi>m<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<div id=\"Figure_02_02_005\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_005\"><span id=\"fs-id1388109\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\" width=\"463\" height=\"370\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/9d1fa5b48ee8553cb85ee038851cc3d8ab61d35b\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">5<\/span>\r\n<span class=\"os-caption\"> Perpendicular lines<\/span><\/div>\r\n<\/div>\r\n<div id=\"Example_02_02_13\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">14<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2496716\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2496718\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither<\/h3>\r\n<p id=\"fs-id2390494\">Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2931411\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2931413\">The first thing we want to do is rewrite the equations so that both equations are in slope-intercept form.<\/p>\r\n<p id=\"fs-id2387008\">First equation:<\/p>\r\n\r\n<div id=\"fs-id1715280\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22124<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22124<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2876107\">Second equation:<\/p>\r\n\r\n<div id=\"fs-id2892807\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>8<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>\u22124<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mo><\/mo><mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>-<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>8<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>\u22124<\/mn><mi>y<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mo><\/mo><mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>-<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2293557\">See the graph of both lines in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_006\">Figure 6<\/a><\/p>\r\n\r\n<div id=\"Figure_02_02_006\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_006\"><span id=\"fs-id1712745\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 4 to 5 and the y-axis ranging from negative 4 to 4. Two functions are graphed on the same plot: y = negative 4 times x\/3 plus 1 and y = 3 times x\/4 minus 2. A box is placed at the intersection to note that it forms a right angle.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 4 to 5 and the y-axis ranging from negative 4 to 4. Two functions are graphed on the same plot: y = negative 4 times x\/3 plus 1 and y = 3 times x\/4 minus 2. A box is placed at the intersection to note that it forms a right angle.\" width=\"361\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7e7df2ee58c838563b2b9ece5e47bc61b00e89cf\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">6<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id2433538\">From the graph, we can see that the lines appear perpendicular, but we must compare the slopes.<\/p>\r\n\r\n<div id=\"fs-id1811234\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>\u22c5<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msub>\r\n<mi>m<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>\u22c5<\/mo><msub>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2699630\">The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2514142\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#11<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_11\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1228195\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1228196\">Graph the two lines and determine whether they are parallel, perpendicular, or neither:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1743214\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Writing the Equations of Lines Parallel or Perpendicular to a Given Line<\/h2>\r\n<p id=\"fs-id1961505\">As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">point-slope formula<\/span> to write the equation of the new line.<\/p>\r\n\r\n<div id=\"fs-id2640149\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2495575\"><strong>Given an equation for a line, write the equation of a line parallel or perpendicular to it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2495579\" type=\"1\">\r\n \t<li>Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept form.<\/li>\r\n \t<li>Use the slope and the given point with the point-slope formula.<\/li>\r\n \t<li>Simplify the line to slope-intercept form and compare the equation to the given line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_14\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">15<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1293659\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2372275\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point<\/h3>\r\n<p id=\"fs-id1803338\">Write the equation of line parallel to a<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1531337\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1531339\">First, we will write the equation in slope-intercept form to find the slope.<\/p>\r\n\r\n<div id=\"fs-id2980407\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u20135<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u20135<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1280284\">The slope is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The <em data-effect=\"italics\">y-<\/em>intercept is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>but that really does not enter into our problem, as the only thing we need for two lines to be parallel is the same slope. The one exception is that if the <em data-effect=\"italics\">y-<\/em>intercepts are the same, then the two lines are the same line. The next step is to use this slope and the given point with the point-slope formula.\r\n<div id=\"fs-id1143286\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2441363\">The equation of the line is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>10.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>10.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_008\">Figure 7<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_02_008\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_02_008\"><span id=\"fs-id2377310\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\" width=\"361\" height=\"376\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/a6abcfdea9f2805a3d47b13fc8c43116ecabe5b6\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">7<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2959024\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#12<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_12\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1717634\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1717636\">Find the equation of the line parallel to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_02_15\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">16<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1223672\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1223674\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point<\/h3>\r\n<p id=\"fs-id1931042\">Find the equation of the line perpendicular to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/annotation-xml><\/semantics><\/math>and passing through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mo>(<\/mo>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><mn>1<\/mn>\r\n<mo>)<\/mo><mo>.<\/mo>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>.<\/mo><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1790301\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1790303\">The first step is to write the equation in slope-intercept form.<\/p>\r\n\r\n<div id=\"fs-id1164124\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><mi>y<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2508746\">We see that the slope is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>m<\/mi><mo>=<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>This means that the slope of the line perpendicular to the given line is the negative reciprocal, or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Next, we use the point-slope formula with this new slope and the given point.\r\n<div id=\"fs-id1824969\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>y<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2308224\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1786321\">Access these online resources for additional instruction and practice with linear equations.<\/p>\r\n\r\n<ul id=\"fs-id1786324\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/rationaleqs\" target=\"_blank\" rel=\"noopener nofollow\">Solving rational equations<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/twopointsline\" target=\"_blank\" rel=\"noopener nofollow\">Equation of a line given two points<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/findperpline\" target=\"_blank\" rel=\"noopener nofollow\">Finding the equation of a line perpendicular to another line through a given point<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/findparaline\" target=\"_blank\" rel=\"noopener nofollow\">Finding the equation of a line parallel to another line through a given point<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.2 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id3264330\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id2508880\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id2508886\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3107034\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2508886-solution\">1<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3107036\">What does it mean when we say that two lines are parallel?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2377547\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2377548\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2377549\">What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2499477\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2499478\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2499477-solution\">3<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2499479\">How do we recognize when an equation, for example<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>will be a straight line (linear) when graphed?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1261499\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2769996\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2769997\">What does it mean when we say that a linear equation is inconsistent?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2770000\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2770001\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2770000-solution\">5<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2438687\">When solving the following equation:<\/p>\r\n<p id=\"fs-id2438690\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n<p id=\"fs-id2794990\">explain why we must exclude<\/p>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nas possible solutions from the solution set.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2497234\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id2020786\">For the following exercises, solve the equation for<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1540386\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1540387\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2019698\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2019700\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2019698-solution\">7<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2400227\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2400228\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3120624\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3120625\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3120624-solution\">9<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1570246\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1570247\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1570248\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1833287\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1833288\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1833287-solution\">11<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1833289\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2371482\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2371483\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2522563\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<mn>6<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<mn>6<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2483678\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2483680\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2483678-solution\">13<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2315466\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2315467\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2315468\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>21<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>21<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2301811\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2301812\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2301811-solution\">15<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2505111\">For the following exercises, solve each rational equation for<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State all <em data-effect=\"italics\">x<\/em>-values that are excluded from the solution set.\r\n<div id=\"fs-id1186641\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1186642\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1186643\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1354301\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2655308\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1354301-solution\">17<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2655309\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1929983\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2653559\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2653560\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>7<\/mn>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2771146\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2771148\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2771146-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2771149\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2266250\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2266251\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2266252\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2505172\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2505173\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2505172-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2933151\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\u00a0<\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2931117\">For the following exercises, find the equation of the line using the point-slope formula.\r\nWrite all the final equations using the slope-intercept form.<\/p>\r\n\r\n<div id=\"fs-id2879070\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2879071\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nwith a slope of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1956932\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1956933\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1956932-solution\">23<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nwith a slope of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>4<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>4<\/mn><\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2020098\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2020099\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2020100\"><em data-effect=\"italics\">x<\/em>-intercept is 1, and<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1322919\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1322920\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1322919-solution\">25<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1388261\"><em data-effect=\"italics\">y<\/em>-intercept is 2, and<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2766380\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2683570\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>,<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>,<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2302767\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1687202\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2302767-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1760345\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2722754\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2722755\">parallel to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and passes through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1928538\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2698648\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1928538-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2698649\">perpendicular to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtext>3<\/mtext><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtext>3<\/mtext><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and passes through the point\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1973853\">For the following exercises, find the equation of the line using the given information.<\/p>\r\n\r\n<div id=\"fs-id1712932\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1712933\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2433146\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2433147\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2433146-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2413598\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2413599\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2413600\">The slope is undefined and it passes through the point<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1337290\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1337291\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1337290-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1337292\">The slope equals zero and it passes through the point<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>\u22124<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>\u22124<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3008660\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3008661\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3008662\">The\u00a0slope\u00a0is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac><\/annotation-xml><\/semantics><\/math>and\u00a0it\u00a0passes\u00a0through\u00a0the\u00a0point\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1545608\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1545609\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1545608-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>\u20131<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>\u20131<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>4<\/mn><mo>,<\/mo><mn>\u20135<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>4<\/mn><mo>,<\/mo><mn>\u20135<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id768131\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id2698028\">For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.<\/p>\r\n\r\n<div id=\"fs-id1719410\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1719412\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>1<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>1<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2515795\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2515797\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2515795-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2515798\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\u00a0<\/mtable><\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\u00a0<\/mtable><\/mrow><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2736413\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1558245\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1558246\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/semantics><\/math><\/p>\r\n\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n\r\n\r\n<annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\u00a0<\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/annotation-xml>\r\n\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1942944\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1942945\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1942944-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1942946\"><math display=\"inline\"><semantics><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mi>y<\/mi><mo>=<\/mo><mn>\u22123<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\r\n<annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mi>y<\/mi><mo>=<\/mo><mn>\u22123<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1700245\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1736475\">For the following exercises, find the slope of the line that passes through the given points.<\/p>\r\n\r\n<div id=\"fs-id1736478\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1736479\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>7<\/mn><mo>,<\/mo><mn>9<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>7<\/mn><mo>,<\/mo><mn>9<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1773932\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1895275\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1773932-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>\u22127<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mn>\u22127<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1846635\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1846636\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3160370\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3160371\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3160370-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2416770\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2416771\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1386040\">For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.<\/p>\r\n\r\n<div id=\"fs-id3165030\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3165031\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3165030-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3165032\"><math display=\"inline\"><semantics><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\r\n<annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2381892\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2381893\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2381894\"><math display=\"inline\"><semantics><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\r\n<annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/annotation-xml>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2431191\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id2699691\">For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the <em data-effect=\"italics\">y<\/em>-intercept occurs. State your ymin and ymax values.<\/p>\r\n\r\n<div id=\"fs-id1390855\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1390856\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1390855-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0.537<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.19<\/mn><mi>y<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0.537<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.19<\/mn><mi>y<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2006746\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2006747\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4,500<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>200<\/mn><mi>y<\/mi><mo>=<\/mo><mn>9,528<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4,500<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>200<\/mn><mi>y<\/mi><mo>=<\/mo><mn>9,528<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2653806\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2653807\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2653806-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>200<\/mn><mo>\u2212<\/mo><mn>30<\/mn><mi>y<\/mi>\r\n<\/mrow>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>70<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>200<\/mn><mo>\u2212<\/mo><mn>30<\/mn><mi>y<\/mi>\r\n<\/mrow>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>70<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2294054\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id2387795\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2387796\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2387797\">Starting with the point-slope formula<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mi>m<\/mi><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>)<\/mo><mo>,<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mi>m<\/mi><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>)<\/mo><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>solve this expression for\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>x<\/mi><\/annotation-xml><\/semantics><\/math>in terms of\r\n\r\n<math display=\"inline\"><semantics><mrow><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>y<\/mi><mo>,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>y<\/mi><mo>,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>m<\/mi><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2683690\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2683691\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2683690-solution\">51<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2683692\">Starting with the standard form of an equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi><\/annotation-xml><\/semantics><\/math>solve this expression for\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><\/annotation-xml><\/semantics><\/math>in terms of\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>A<\/mi><mo>,<\/mo><mi>B<\/mi><mo>,<\/mo><mi>C<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>A<\/mi><mo>,<\/mo><mi>B<\/mi><mo>,<\/mo><mi>C<\/mi><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>x<\/mi><\/annotation-xml><\/semantics><\/math>. Then put the expression in slope-intercept form.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2638987\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2638988\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2638989\">Use the above derived formula to put the following standard equation in slope intercept form:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>=<\/mo><mn>25.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>=<\/mo><mn>25.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2519655\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2519656\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2519655-solution\">53<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2519657\">Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.<\/p>\r\n<math display=\"inline\"><semantics><mrow><mo>(<\/mo><mo>\u2013<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mo>\u2013<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>0<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>0<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1972709\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1972710\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1972711\">Find the slopes of the diagonals in the previous exercise. Are they perpendicular?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1570540\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1274216\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1274217\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1274216-solution\">55<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1274218\">The slope for a wheelchair ramp for a home has to be<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.\r\n\r\n<span id=\"fs-id1471955\" data-type=\"media\" data-alt=\"\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"\" width=\"402\" height=\"86\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/bded289bae0a02e51a163adfd8db52d5c7df5cc9\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3040342\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3040343\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3040344\">If the profit equation for a small business selling<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>number of item one and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>number of item two is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>find the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>value when\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mtext>$<\/mtext><mn>453<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0\u00a0<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>75.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mtext>$<\/mtext><mn>453<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0\u00a0<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>75.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id3231640\">For the following exercises, use this scenario: The cost of renting a car is $45\/wk plus $0.25\/mi traveled during that week. An equation to represent the cost would be<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>45<\/mn><mo>+<\/mo><mn>.25<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>45<\/mn><mo>+<\/mo><mn>.25<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>is the number of miles traveled.\r\n<div id=\"fs-id3148386\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3148387\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3148386-solution\">57<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3148388\">What is your cost if you travel 50 mi?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1273127\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1273128\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1273129\">If your cost were<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtext>$<\/mtext><mn>63.75<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtext>$<\/mtext><mn>63.75<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>how many miles were you charged for traveling?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1315527\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1315528\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1315527-solution\">59<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1315529\">Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_eb6fde5e-763a-412f-8aa4-866649f706d5\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"ui-has-child-title\" data-type=\"abstract\">\n<header>\n<h2 data-type=\"title\">Learning Objectives<\/h2>\n<\/header>\n<section>\n<p id=\"para-00001\">In this section, you will:<\/p>\n<ul id=\"list-00001\">\n<li>Solve equations in one variable algebraically.<\/li>\n<li>Solve a rational equation.<\/li>\n<li>Find a linear equation.<\/li>\n<li>Given the equations of two lines, determine whether their graphs are parallel or perpendicular.<\/li>\n<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p id=\"fs-id1402487\">Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00\/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_001\">Figure 1<\/a>.<\/p>\n<div id=\"Figure_02_02_001\" class=\"os-figure\">\n<figure class=\"medium\" data-id=\"Figure_02_02_001\"><span id=\"fs-id1294390\" data-type=\"media\" data-alt=\"Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).\" width=\"509\" height=\"352\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/5e03adc287df01ae9cc9debe0416a2a49351b431\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<section id=\"fs-id2454936\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Linear Equations in One Variable<\/h2>\n<p id=\"fs-id1325030\">A <span id=\"term-00001\" data-type=\"term\">linear equation<\/span> is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and are solved using basic algebraic operations.<\/p>\n<p id=\"fs-id2440186\">We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An <span id=\"term-00002\" data-type=\"term\">identity equation<\/span> is true for all values of the variable. Here is an example of an identity equation.<\/p>\n<div id=\"fs-id1384930\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1472975\">The <span id=\"term-00003\" data-type=\"term\">solution set<\/span> consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>will make the equation true.<\/p>\n<p id=\"fs-id2906254\">A <span id=\"term-00004\" data-type=\"term\">conditional equation<\/span> is true for only some values of the variable. For example, if we are to solve the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we have the following:<\/p>\n<div id=\"fs-id1332285\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1518040\">The solution set consists of one number:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>It is the only solution and, therefore, we have solved a conditional equation.<\/p>\n<p id=\"fs-id1476044\">An <span id=\"term-00005\" data-type=\"term\">inconsistent equation<\/span> results in a false statement. For example, if we are to solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>5<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>5<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we have the following:<\/p>\n<div id=\"fs-id722242\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><mn>5<\/mn><mi>x<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>\u2260<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221220<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>False\u00a0statement<\/mtext><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><mn>5<\/mn><mi>x<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>\u2260<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221220<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>False\u00a0statement<\/mtext><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1554254\">Indeed,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u221215<\/mn><mo>\u2260<\/mo><mspace width=\"0.5em\"><\/mspace><mn>\u221220.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u221215<\/mn><mo>\u2260<\/mo><mspace width=\"0.5em\"><\/mspace><mn>\u221220.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There is no solution because this is an inconsistent equation.<\/p>\n<p id=\"fs-id2279267\">Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.<\/p>\n<div id=\"fs-id1549902\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Linear Equation in One Variable<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1943551\">A linear equation in one variable can be written in the form<\/p>\n<div id=\"fs-id2519610\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1511576\">where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b <\/em>are real numbers,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1592386\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1337817\"><strong>Given a linear equation in one variable, use algebra to solve it.<\/strong><\/p>\n<p id=\"fs-id1759614\">The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mo>_________,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mo>_________,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>if <em data-effect=\"italics\">x <\/em>is the unknown. There is no set order, as the steps used depend on what is given:<\/p>\n<ol id=\"fs-id1334010\" type=\"1\">\n<li>We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.<\/li>\n<li>Apply the distributive property as needed:<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Isolate the variable on one side of the equation.<\/li>\n<li>When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_01\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id3264315\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1294630\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Equation in One Variable<\/h3>\n<p id=\"fs-id1778894\">Solve the following equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>19.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>19.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id780975\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1215275\">This equation can be written in the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>by subtracting<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>19<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>19<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>from both sides. However, we may proceed to solve the equation in its original form by performing algebraic operations.<\/p>\n<div id=\"fs-id1429109\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>19<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a07\u00a0from\u00a0both\u00a0sides<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0divide\u00a0by\u00a02<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>19<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a07\u00a0from\u00a0both\u00a0sides<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0divide\u00a0by\u00a02<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1793429\">The solution is 6.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1531582\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2980056\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1954826\">Solve the linear equation in one variable:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>\u22129.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>\u22129.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_02\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1542084\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2439455\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Equation Algebraically When the Variable Appears on Both Sides<\/h3>\n<p id=\"fs-id2726928\">Solve the following equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mn>12<\/mn><mo>=<\/mo><mn>15<\/mn><mn>\u22125<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mn>12<\/mn><mo>=<\/mo><mn>15<\/mn><mn>\u22125<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2439343\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1943089\">Apply standard algebraic properties.<\/p>\n<div id=\"fs-id2443828\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Apply\u00a0the\u00a0distributive\u00a0property<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>9<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Place\u00a0<\/mtext><mi>x<\/mi><mtext>-terms\u00a0on\u00a0one\u00a0side\u00a0and\u00a0simplify<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mrow><mn>15<\/mn><\/mrow><mn>9<\/mn><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac>\n<mn>1<\/mn>\n<mn>9<\/mn>\n<\/mfrac>\n<mtext>,\u00a0the\u00a0reciprocal\u00a0of\u00a09<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Apply\u00a0the\u00a0distributive\u00a0property<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>9<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221215<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Place\u00a0<\/mtext><mi>x<\/mi><mtext>-terms\u00a0on\u00a0one\u00a0side\u00a0and\u00a0simplify<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mrow><mn>15<\/mn><\/mrow><mn>9<\/mn><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0both\u00a0sides\u00a0by\u00a0<\/mtext><mfrac>\n<mn>1<\/mn>\n<mn>9<\/mn>\n<\/mfrac>\n<mtext>,\u00a0the\u00a0reciprocal\u00a0of\u00a09<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1333518\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\n<p id=\"fs-id1469028\">This problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach the final line,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1533557\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2439622\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1287709\">Solve the equation in one variable:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1352666\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving a Rational Equation<\/h2>\n<p id=\"fs-id2697408\">In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a <strong>rational equation<\/strong>.<\/p>\n<p id=\"fs-id1568247\">Recall that a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">rational number<\/span> is the ratio of two numbers, such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>A <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">rational expression<\/span> is the ratio, or quotient, of two polynomials. Here are three examples.<\/p>\n<div id=\"fs-id2503300\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mfrac>\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mfrac>\n<mn>4<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mfrac>\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mfrac>\n<mn>4<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2507716\">Rational equations have a variable in the denominator in at least one of the terms.<br \/>\nOur goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">least common denominator<\/span> (LCD).<\/p>\n<p id=\"fs-id1290385\">Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.<\/p>\n<div id=\"Example_02_02_03\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2707414\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1275451\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Rational Equation<\/h3>\n<p id=\"fs-id2500534\">Solve the rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>22<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>22<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2364839\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2708670\">We have three denominators;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and 3. The LCD must contain<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>,<\/mo><mn>3<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and 3. An LCD of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>contains all three denominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the equation by the LCD<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1149071\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Use\u00a0the\u00a0distributive\u00a0property<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>3<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mn>6<\/mn>\n<\/menclose>\n<mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Cancel\u00a0out\u00a0the\u00a0common\u00a0factors<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0remaining\u00a0factors\u00a0by\u00a0each\u00a0numerator<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>21<\/mn><mo>\u2212<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>11<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mrow><mn>11<\/mn><\/mrow><mrow><mn>44<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Use\u00a0the\u00a0distributive\u00a0property<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>6<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>3<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><menclose notation=\"updiagonalstrike\">\n<mn>6<\/mn>\n<\/menclose>\n<mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Cancel\u00a0out\u00a0the\u00a0common\u00a0factors<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply\u00a0remaining\u00a0factors\u00a0by\u00a0each\u00a0numerator<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>21<\/mn><mo>\u2212<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>11<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>44<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mrow><mn>11<\/mn><\/mrow><mrow><mn>44<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2382754\">A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial\u2014two terms added or subtracted\u2014such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Always consider a binomial as an individual factor\u2014the terms cannot be separated. For example, suppose a problem has three terms and the denominators are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>First, factor all denominators. We then have<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>The<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>in the first denominator is separate from the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>in the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>one factor of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the 3. Thus, the LCD is the following:<\/p>\n<div id=\"fs-id2268029\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1510962\">So, both sides of the equation would be multiplied by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.<\/p>\n<p id=\"fs-id1780189\">Another example is a problem with two denominators, such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Once the second denominator is factored as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>there is a common factor of <em data-effect=\"italics\">x<\/em> in both denominators and the LCD is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1829569\">Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.<\/p>\n<div id=\"fs-id1467328\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mi>a<\/mi>\n<mi>b<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>d<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mi>a<\/mi>\n<mi>b<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>d<\/mi>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id914245\">We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.<\/p>\n<p><span id=\"fs-id2119795\" data-type=\"media\" data-alt=\"\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"\" width=\"487\" height=\"43\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7d9d5ac2f0af14cb4dab1f5490e09fe724be0e36\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id2639399\">Multiply<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mrow><mo>(<\/mo>\n<mi>d<\/mi>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mrow><mo>(<\/mo>\n<mi>d<\/mi>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mrow><mo>(<\/mo>\n<mi>c<\/mi>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mrow><mo>(<\/mo>\n<mi>c<\/mi>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>which results in<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>d<\/mi><mo>=<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>d<\/mi><mo>=<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1551812\">Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.<\/p>\n<div id=\"fs-id1931811\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Rational Equations<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2629095\">A <span id=\"term-00009\" data-type=\"term\">rational equation<\/span> contains at least one rational expression where the variable appears in at least one of the denominators.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1147894\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1519809\"><strong>Given a rational equation, solve it.<\/strong><\/p>\n<ol id=\"fs-id2667782\" type=\"1\">\n<li>Factor all denominators in the equation.<\/li>\n<li>Find and exclude values that set each denominator equal to zero.<\/li>\n<li>Find the LCD.<\/li>\n<li>Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left.<\/li>\n<li>Solve the remaining equation.<\/li>\n<li>Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_04\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1831988\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1504300\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Rational Equation without Factoring<\/h3>\n<p id=\"fs-id2435055\">Solve the following rational equation:<\/p>\n<div id=\"fs-id863002\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>7<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/div>\n<details id=\"fs-id2502004\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id3141796\">We have three denominators:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>No factoring is required. The product of the first two denominators is equal to the third denominator, so, the LCD is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Only one value is excluded from a solution set, 0.<\/p>\n<p><math display=\"inline\"><semantics><mrow>&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>Next, multiply the whole equation (both sides of the equal sign) by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2503271\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>2<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd><mrow><mrow><mo>(<\/mo> <mrow><mfrac><mn>7<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mi>x<\/mi>\n<\/menclose>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mi>x<\/mi>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0<\/mtext><mn>2<\/mn><mi>x<\/mi><mtext>.<\/mtext><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Denominators\u00a0cancel\u00a0out<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22123<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>3<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mtext>or<\/mtext><mspace width=\"0.3em\"><\/mspace><mrow><mo>{<\/mo> <mrow>\n<mn>\u22121<\/mn><\/mrow> <mo>}<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>2<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd><mrow><mrow><mo>(<\/mo> <mrow><mfrac><mn>7<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mi>x<\/mi>\n<\/menclose>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mi>x<\/mi>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0<\/mtext><mn>2<\/mn><mi>x<\/mi><mtext>.<\/mtext><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Denominators\u00a0cancel\u00a0out<\/mtext><mo>.<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>4<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>7<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22123<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>3<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mtext>or<\/mtext><mspace width=\"0.3em\"><\/mspace><mrow><mo>{<\/mo> <mrow>\n<mn>\u22121<\/mn><\/mrow> <mo>}<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1496122\">The proposed solution is \u22121,<\/p>\n<p><math display=\"inline\"><semantics><mrow>&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>which is not an excluded value, so the solution set contains one number,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn> \u22121<\/mn><mo>, <\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn> \u22121<\/mn><mo>, <\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mn>\u22121<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mn>\u22121<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>written in set notation.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1254485\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1476079\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1476080\">Solve the rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>6<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>6<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_05\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1533182\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1595938\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Rational Equation by Factoring the Denominator<\/h3>\n<p id=\"fs-id3268868\">Solve the following rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1204080\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2436750\">First find the common denominator. The three denominators in factored form are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>,<\/mo><mn>10<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>,<\/mo><mn>10<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The smallest expression that is divisible by each one of the denominators is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Only<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>is an excluded value. Multiply the whole equation by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>20<\/mn><mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1581638\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>20<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mn>20<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>20<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>35<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>35<\/mn><\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>20<\/mn><mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mn>20<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>20<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>35<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>35<\/mn><\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2437669\">The solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1844482\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1768288\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1768289\">Solve the rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_06\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1536446\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2682405\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving Rational Equations with a Binomial in the Denominator<\/h3>\n<p id=\"fs-id1723264\">Solve the following rational equations and state the excluded values:<\/p>\n<ol id=\"fs-id1500286\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mi>x<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mi>x<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id1216224\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<ol id=\"fs-id1996554\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span>\n<p id=\"fs-id833086\">The denominators<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>have nothing in common. Therefore, the LCD is the product<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>However, for this problem, we can cross-multiply.<\/p>\n<div id=\"fs-id1447286\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mn>3<\/mn><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mfrac><mn>5<\/mn><mi>x<\/mi><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Distribute<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22122<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221230<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mfrac><mn>3<\/mn><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mfrac><mn>5<\/mn><mi>x<\/mi><\/mfrac><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mspace width=\"2em\"><\/mspace><mtext>Distribute<\/mtext><mtext>.<\/mtext><\/mrow><\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>30<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mn>\u22122<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u221230<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2493985\">The solution is 15.<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><\/annotation-xml><\/semantics><\/math>The excluded values are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<li><span class=\"token\">\u24d1<\/span>\n<p id=\"fs-id2434774\">The LCD is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply both sides of the equation by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1901302\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/menclose>\n<mi>x<\/mi><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/menclose>\n<mn>5<\/mn><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>13<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/menclose>\n<mi>x<\/mi><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><menclose notation=\"updiagonalstrike\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/menclose>\n<mn>5<\/mn><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<menclose notation=\"updiagonalstrike\">\n<mn>2<\/mn>\n<\/menclose>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>13<\/mn><\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1783284\">The solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>13<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>13<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The excluded value is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<li><span class=\"token\">\u24d2<\/span>\n<p id=\"fs-id3156711\">The least common denominator is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply both sides of the equation by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1289204\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>12<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mi>x<\/mi>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>12<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1923900\">The solution is 4. The excluded value is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1686117\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2869362\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2869363\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State the excluded values.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_07\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1823223\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1387285\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Rational Equation with Factored Denominators and Stating Excluded Values<\/h3>\n<p id=\"fs-id3094632\">Solve the rational equation after factoring the denominators:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State the excluded values.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1527395\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2437930\">We must factor the denominator<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22121.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22121.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We recognize this as the difference of squares, and factor it as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Thus, the LCD that contains each denominator is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.<\/p>\n<div id=\"fs-id1958485\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0negative\u00a0sign<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0negative\u00a0sign<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1847437\">The solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>The excluded values are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>1<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>1<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1475570\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2507037\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1425920\">Solve the rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2673660\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding a Linear Equation<\/h2>\n<p id=\"fs-id2803187\">Perhaps the most familiar form of a linear equation is the slope-intercept form, written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mtext>slope<\/mtext> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mtext>slope<\/mtext> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mo>=<\/mo><mi>y<\/mi><mtext>-intercept<\/mtext><mtext>.<\/mtext> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mo>=<\/mo><mi>y<\/mi><mtext>-intercept<\/mtext><mtext>.<\/mtext> <\/mrow><\/annotation-xml><\/semantics><\/math>Let us begin with the slope.<\/p>\n<section id=\"fs-id2737079\" data-depth=\"2\">\n<h3 data-type=\"title\">The Slope of a Line<\/h3>\n<p id=\"fs-id1539119\">The <span id=\"term-00010\" data-type=\"term\">slope<\/span> of a line refers to the ratio of the vertical change in <em data-effect=\"italics\">y<\/em> over the horizontal change in <em data-effect=\"italics\">x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n<div id=\"fs-id1920678\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1525950\">If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_002\">Figure 2<\/a>. The lines indicate the following slopes:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"Figure_02_02_002\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_002\"><span id=\"fs-id1939377\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/143ef80bb95f29cd295ad08addad53ad6864937a\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/div>\n<\/div>\n<div id=\"fs-id1467947\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Slope of a Line<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2512513\">The slope of a line, <em data-effect=\"italics\">m<\/em>, represents the change in <em data-effect=\"italics\">y<\/em> over the change in <em data-effect=\"italics\">x.<\/em> Given two points,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>the following formula determines the slope of a line containing these points:<\/p>\n<div id=\"fs-id1908930\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_08\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2388781\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2388784\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Slope of a Line Given Two Points<\/h3>\n<p id=\"fs-id2952969\">Find the slope of a line that passes through the points<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2715020\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2715022\">We substitute the <em data-effect=\"italics\">y-<\/em>values and the <em data-effect=\"italics\">x-<\/em>values into the formula.<\/p>\n<div id=\"fs-id3234414\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>\u22125<\/mn><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mrow>\n<mn>\u22127<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>7<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>\u22125<\/mn><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mrow>\n<mn>\u22127<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>7<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1568234\">The slope is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>7<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>7<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1560357\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\n<p id=\"fs-id1477668\">It does not matter which point is called<\/p>\n<p id=\"fs-id1477666\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub><\/mrow><\/mrow><\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>)<\/p>\n<p>(<\/p>\n<p>x<br \/>\n1<\/p>\n<p>,<br \/>\ny<br \/>\n1<\/p>\n<p>)<\/p>\n<p>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>2<\/mn>\n<\/msub><\/mrow><\/mrow><\/mrow><\/mrow><\/semantics><\/math>).<\/p>\n<p>(<\/p>\n<p>x<br \/>\n2<\/p>\n<p>,<br \/>\ny<br \/>\n2<\/p>\n<p>).<\/p>\n<p>As long as we are consistent with the order of the <em data-effect=\"italics\">y<\/em> terms and the order of the <em data-effect=\"italics\">x<\/em> terms in the numerator and denominator, the calculation will yield the same result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1225501\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1553581\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1482316\">Find the slope of the line that passes through the points<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_09\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1424004\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1581998\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Identifying the Slope and <em data-effect=\"italics\">y-<\/em>intercept of a Line Given an Equation<\/h3>\n<p id=\"fs-id1386856\">Identify the slope and <em data-effect=\"italics\">y-<\/em>intercept, given the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2517224\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1763671\">As the line is in<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>form, the given line has a slope of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The <em data-effect=\"italics\">y-<\/em>intercept is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>\u22124.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>\u22124.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1918796\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\n<p id=\"eip-id1266666\">The <em data-effect=\"italics\">y<\/em>-intercept is the point at which the line crosses the <em data-effect=\"italics\">y-<\/em>axis. On the <em data-effect=\"italics\">y-<\/em>axis,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We can always identify the <em data-effect=\"italics\">y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em data-effect=\"italics\">b.<\/em> Or, just substitute<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>0<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and solve for <em data-effect=\"italics\">y.<\/em><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1520430\" data-depth=\"2\">\n<h3 data-type=\"title\">The Point-Slope Formula<\/h3>\n<p id=\"fs-id1560189\">Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.<\/p>\n<div id=\"fs-id1539802\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2497497\">This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.<\/p>\n<div id=\"fs-id1401850\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Point-Slope Formula<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1926574\">Given one point and the slope, the point-slope formula will lead to the equation of a line:<\/p>\n<div id=\"fs-id1939371\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mi>m<\/mi><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_09a\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1780029\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1780031\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Equation of a Line Given the Slope and One Point<\/h3>\n<p id=\"fs-id1517411\">Write the equation of the line with slope<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the final equation in slope-intercept form.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1557465\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1562535\">Using the point-slope formula, substitute<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>for <em data-effect=\"italics\">m <\/em>and the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>8<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1537532\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>20<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>20<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id2736677\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\n<p id=\"fs-id2410998\">Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1939337\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1814119\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1814120\">Given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>find the equation of the line in slope-intercept form passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_10\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">11<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2279286\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2694098\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Equation of a Line Passing Through Two Given Points<\/h3>\n<p id=\"fs-id1517573\">Find the equation of the line passing through the points<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the final equation in slope-intercept form.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1920249\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1920251\">First, we calculate the slope using the slope formula and two points.<\/p>\n<div id=\"fs-id1786245\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1539521\">Next, we use the point-slope formula with the slope of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and either point. Let\u2019s pick the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1572614\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0<\/mtext><mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mspace width=\"2em\"><\/mspace><mtext>Distribute\u00a0the\u00a0<\/mtext><mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1914286\">In slope-intercept form, the equation is written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1592294\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h3>\n<p id=\"fs-id1940795\">To prove that either point can be used, let us use the second point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and see if we get the same equation.<\/p>\n<div id=\"fs-id1790167\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1562421\">We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2520904\" data-depth=\"2\">\n<h3 data-type=\"title\">Standard Form of a Line<\/h3>\n<p id=\"fs-id2508791\">Another way that we can represent the equation of a line is in <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">standard form<\/span>. Standard form is given as<\/p>\n<div id=\"fs-id3150952\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1422421\">where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>A<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>A<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>B<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>B<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>C<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>C<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>are integers. The <em data-effect=\"italics\">x- <\/em>and <em data-effect=\"italics\">y-<\/em>terms are on one side of the equal sign and the constant term is on the other side.<\/p>\n<div id=\"Example_02_02_11\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">12<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2431257\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2431259\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Equation of a Line and Writing It in Standard Form<\/h3>\n<p id=\"fs-id2736527\">Find the equation of the line with<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22126<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>\u22126<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the equation in standard form.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1388008\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2753839\">We begin using the point-slope formula.<\/p>\n<div id=\"fs-id1353048\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22126<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22126<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1357955\">From here, we multiply through by 2, as no fractions are permitted in standard form, and then move both variables to the left aside of the equal sign and move the constants to the right.<\/p>\n<div id=\"fs-id1315939\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22126<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1385368\">This equation is now written in standard form.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2426862\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2820293\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2820294\">Find the equation of the line in standard form with slope<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1548365\" data-depth=\"2\">\n<h3 data-type=\"title\">Vertical and Horizontal Lines<\/h3>\n<p id=\"fs-id3207599\">The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">vertical line<\/span> is given as<\/p>\n<div id=\"fs-id2002088\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2386398\">where <em data-effect=\"italics\">c <\/em>is a constant. The slope of a vertical line is undefined, and regardless of the <em data-effect=\"italics\">y-<\/em>value of any point on the line, the <em data-effect=\"italics\">x-<\/em>coordinate of the point will be <em data-effect=\"italics\">c<\/em>.<\/p>\n<p id=\"fs-id1569566\">Suppose that we want to find the equation of a line containing the following points:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>\u22125<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>\u22125<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>First, we will find the slope.<\/p>\n<div id=\"fs-id1400578\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>0<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>0<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1709161\">Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the <em data-effect=\"italics\">x-<\/em>coordinates are the same and we find a vertical line through<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>\u22123.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>\u22123.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_003\">Figure 3<\/a><strong>.<\/strong><\/p>\n<p id=\"fs-id2503257\">The equation of a <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">horizontal line<\/span> is given as<\/p>\n<div id=\"fs-id1477479\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1513867\">where <em data-effect=\"italics\">c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em data-effect=\"italics\">x-<\/em>value of a point on the line, the <em data-effect=\"italics\">y-<\/em>coordinate will be <em data-effect=\"italics\">c<\/em>.<\/p>\n<p id=\"fs-id2528940\">Suppose we want to find the equation of a line that contains the following set of points:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>We can use the point-slope formula. First, we find the slope using any two points on the line.<\/p>\n<div id=\"fs-id1257613\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>0<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>m<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>0<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1258397\">Use any point for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msub>\n<mi>x<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>,<\/mo><msub>\n<mi>y<\/mi>\n<mn>1<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>in the formula, or use the <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<div id=\"fs-id1691149\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>0<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mi>y<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>0<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3257368\">The graph is a horizontal line through<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>\u22122.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>\u22122.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>Notice that all of the <em data-effect=\"italics\">y-<\/em>coordinates are the same. See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_003\">Figure 3<\/a>.<\/p>\n<div id=\"Figure_02_02_003\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_003\"><span id=\"fs-id1336884\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/2d21dfa5c23edca204827fda8f8f28b5312737a5\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">3<\/span><br \/>\n<span class=\"os-caption\">The line <em data-effect=\"italics\">x<\/em> = \u22123 is a vertical line. The line <em data-effect=\"italics\">y<\/em> = \u22122 is a horizontal line.<\/span><\/div>\n<\/div>\n<div id=\"Example_02_02_12\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">13<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2797158\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2797160\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Equation of a Line Passing Through the Given Points<\/h3>\n<p id=\"fs-id1273666\">Find the equation of the line passing through the given points:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>\u22123<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>4<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1836962\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2764547\">The <em data-effect=\"italics\">x-<\/em>coordinate of both points is 1. Therefore, we have a vertical line,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1813611\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1538834\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1538835\">Find the equation of the line passing through<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1495944\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Whether Graphs of Lines are Parallel or Perpendicular<\/h2>\n<p id=\"fs-id1973968\">Parallel lines have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts. Lines that are <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">parallel<\/span> to each other will never intersect. For example, <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_004\">Figure 4<\/a> shows the graphs of various lines with the same slope,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>2.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mn>2.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"Figure_02_02_004\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_004\"><span id=\"fs-id1847446\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\" width=\"487\" height=\"593\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/346b578e36322d966a04b3ec8131b4f43419e060\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">4<\/span><br \/>\n<span class=\"os-caption\"> Parallel lines<\/span><\/div>\n<\/div>\n<p id=\"fs-id1929226\">All of the lines shown in the graph are parallel because they have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts.<\/p>\n<p id=\"fs-id1955734\">Lines that are <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">perpendicular<\/span> intersect to form a<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>90\u00b0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>90\u00b0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>-angle. The slope of one line is the negative <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">reciprocal<\/span> of the other. We can show that two lines are perpendicular if the product of the two slopes is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22121<\/mn><mo>:<\/mo><msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>\u22c5<\/mo><msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22121<\/mn><mo>:<\/mo><msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>\u22c5<\/mo><msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>For example, <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_005\">Figure 5<\/a> shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2527184\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><msub><mi>m<\/mi><mn>1<\/mn><\/msub><mo>\u22c5<\/mo><msub><mi>m<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><msub><mi>m<\/mi><mn>1<\/mn><\/msub><mo>\u22c5<\/mo><msub><mi>m<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22121<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<div id=\"Figure_02_02_005\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_005\"><span id=\"fs-id1388109\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\" width=\"463\" height=\"370\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/9d1fa5b48ee8553cb85ee038851cc3d8ab61d35b\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">5<\/span><br \/>\n<span class=\"os-caption\"> Perpendicular lines<\/span><\/div>\n<\/div>\n<div id=\"Example_02_02_13\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">14<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2496716\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2496718\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither<\/h3>\n<p id=\"fs-id2390494\">Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2931411\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2931413\">The first thing we want to do is rewrite the equations so that both equations are in slope-intercept form.<\/p>\n<p id=\"fs-id2387008\">First equation:<\/p>\n<div id=\"fs-id1715280\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22124<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22124<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2876107\">Second equation:<\/p>\n<div id=\"fs-id2892807\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>8<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>\u22124<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mo><\/mo><mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>8<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>\u22124<\/mn><mi>y<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mo><\/mo><mn>\u22123<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>y<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2293557\">See the graph of both lines in <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_006\">Figure 6<\/a><\/p>\n<div id=\"Figure_02_02_006\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_006\"><span id=\"fs-id1712745\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 4 to 5 and the y-axis ranging from negative 4 to 4. Two functions are graphed on the same plot: y = negative 4 times x\/3 plus 1 and y = 3 times x\/4 minus 2. A box is placed at the intersection to note that it forms a right angle.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 4 to 5 and the y-axis ranging from negative 4 to 4. Two functions are graphed on the same plot: y = negative 4 times x\/3 plus 1 and y = 3 times x\/4 minus 2. A box is placed at the intersection to note that it forms a right angle.\" width=\"361\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7e7df2ee58c838563b2b9ece5e47bc61b00e89cf\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">6<\/span><\/div>\n<\/div>\n<p id=\"fs-id2433538\">From the graph, we can see that the lines appear perpendicular, but we must compare the slopes.<\/p>\n<div id=\"fs-id1811234\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>\u22c5<\/mo><msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msub>\n<mi>m<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mo>\u22c5<\/mo><msub>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2699630\">The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2514142\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#11<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_11\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1228195\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1228196\">Graph the two lines and determine whether they are parallel, perpendicular, or neither:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1743214\" data-depth=\"1\">\n<h2 data-type=\"title\">Writing the Equations of Lines Parallel or Perpendicular to a Given Line<\/h2>\n<p id=\"fs-id1961505\">As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">point-slope formula<\/span> to write the equation of the new line.<\/p>\n<div id=\"fs-id2640149\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2495575\"><strong>Given an equation for a line, write the equation of a line parallel or perpendicular to it.<\/strong><\/p>\n<ol id=\"fs-id2495579\" type=\"1\">\n<li>Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept form.<\/li>\n<li>Use the slope and the given point with the point-slope formula.<\/li>\n<li>Simplify the line to slope-intercept form and compare the equation to the given line.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_14\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">15<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1293659\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2372275\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point<\/h3>\n<p id=\"fs-id1803338\">Write the equation of line parallel to a<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1531337\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1531339\">First, we will write the equation in slope-intercept form to find the slope.<\/p>\n<div id=\"fs-id2980407\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u20135<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u20135<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1280284\">The slope is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>The <em data-effect=\"italics\">y-<\/em>intercept is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>but that really does not enter into our problem, as the only thing we need for two lines to be parallel is the same slope. The one exception is that if the <em data-effect=\"italics\">y-<\/em>intercepts are the same, then the two lines are the same line. The next step is to use this slope and the given point with the point-slope formula.<\/p>\n<div id=\"fs-id1143286\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2441363\">The equation of the line is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>10.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>10.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-2-linear-equations-in-one-variable#Figure_02_02_008\">Figure 7<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_02_008\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_02_008\"><span id=\"fs-id2377310\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\" width=\"361\" height=\"376\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/a6abcfdea9f2805a3d47b13fc8c43116ecabe5b6\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">7<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2959024\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#12<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_12\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1717634\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1717636\">Find the equation of the line parallel to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_02_15\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">16<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1223672\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1223674\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point<\/h3>\n<p id=\"fs-id1931042\">Find the equation of the line perpendicular to<\/p>\n<p><math display=\"inline\"><semantics><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/annotation-xml><\/semantics><\/math>and passing through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mo>(<\/mo>\n<mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><mn>1<\/mn>\n<mo>)<\/mo><mo>.<\/mo>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>.<\/mo><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1790301\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1790303\">The first step is to write the equation in slope-intercept form.<\/p>\n<div id=\"fs-id1164124\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22123<\/mn><mi>y<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2508746\">We see that the slope is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>m<\/mi><mo>=<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>This means that the slope of the line perpendicular to the given line is the negative reciprocal, or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Next, we use the point-slope formula with this new slope and the given point.<\/p>\n<div id=\"fs-id1824969\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>y<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>7<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2308224\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1786321\">Access these online resources for additional instruction and practice with linear equations.<\/p>\n<ul id=\"fs-id1786324\">\n<li><a href=\"http:\/\/openstax.org\/l\/rationaleqs\" target=\"_blank\" rel=\"noopener nofollow\">Solving rational equations<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/twopointsline\" target=\"_blank\" rel=\"noopener nofollow\">Equation of a line given two points<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/findperpline\" target=\"_blank\" rel=\"noopener nofollow\">Finding the equation of a line perpendicular to another line through a given point<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/findparaline\" target=\"_blank\" rel=\"noopener nofollow\">Finding the equation of a line parallel to another line through a given point<\/a><\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.2 Section Exercises<\/span><\/h2>\n<section id=\"fs-id3264330\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id2508880\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id2508886\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3107034\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2508886-solution\">1<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3107036\">What does it mean when we say that two lines are parallel?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2377547\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2377548\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2377549\">What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2499477\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2499478\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2499477-solution\">3<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2499479\">How do we recognize when an equation, for example<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>will be a straight line (linear) when graphed?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1261499\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2769996\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2769997\">What does it mean when we say that a linear equation is inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2770000\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2770001\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2770000-solution\">5<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2438687\">When solving the following equation:<\/p>\n<p id=\"fs-id2438690\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>2<\/p>\n<p>x\u22125<\/p>\n<p>=<br \/>\n4<\/p>\n<p>x+1<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id2794990\">explain why we must exclude<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>as possible solutions from the solution set.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2497234\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id2020786\">For the following exercises, solve the equation for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1540386\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1540387\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2019698\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2019700\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2019698-solution\">7<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2400227\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2400228\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3120624\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3120625\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3120624-solution\">9<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1570246\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1570247\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1570248\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>=<\/mo><mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>1<br \/>\n2<\/p>\n<p>\u2212<br \/>\n1<br \/>\n3<\/p>\n<p>x=<br \/>\n4<br \/>\n3<br \/>\n\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1833287\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1833288\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1833287-solution\">11<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1833289\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>x<br \/>\n3<\/p>\n<p>\u2212<br \/>\n3<br \/>\n4<\/p>\n<p>=<\/p>\n<p>2x+3<\/p>\n<p>12<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2371482\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2371483\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2522563\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<mn>6<\/mn>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>2<br \/>\n3<\/p>\n<p>x+<br \/>\n1<br \/>\n2<\/p>\n<p>=<\/p>\n<p>31<\/p>\n<p>6<br \/>\n\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2483678\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2483680\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2483678-solution\">13<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2315466\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2315467\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2315468\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mi>x<\/mi>\n<mn>6<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mn>21<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>2x<\/p>\n<p>3<\/p>\n<p>\u2212<br \/>\n3<br \/>\n4<\/p>\n<p>=<br \/>\nx<br \/>\n6<\/p>\n<p>+<\/p>\n<p>21<\/p>\n<p>4<br \/>\n\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2301811\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2301812\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2301811-solution\">15<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2505111\">For the following exercises, solve each rational equation for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>State all <em data-effect=\"italics\">x<\/em>-values that are excluded from the solution set.<\/p>\n<div id=\"fs-id1186641\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1186642\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1186643\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>3<br \/>\nx<\/p>\n<p>\u2212<br \/>\n1<br \/>\n3<\/p>\n<p>=<br \/>\n1<br \/>\n6<br \/>\n\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1354301\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2655308\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1354301-solution\">17<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2655309\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>2\u2212<br \/>\n3<\/p>\n<p>x+4<\/p>\n<p>=<\/p>\n<p>x+2<\/p>\n<p>x+4<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1929983\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2653559\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2653560\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>7<\/mn>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>3<\/p>\n<p>x\u22122<\/p>\n<p>=<br \/>\n1<\/p>\n<p>x\u22121<\/p>\n<p>+<br \/>\n7<\/p>\n<p>(x\u22121)(x\u22122)<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2771146\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2771148\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2771146-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2771149\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>3x<\/p>\n<p>x\u22121<\/p>\n<p>+2=<br \/>\n3<\/p>\n<p>x\u22121<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2266250\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2266251\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2266252\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>6<\/mn>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>5<\/p>\n<p>x+1<\/p>\n<p>+<br \/>\n1<\/p>\n<p>x\u22123<\/p>\n<p>=<\/p>\n<p>\u22126<\/p>\n<p>x<br \/>\n2<\/p>\n<p>\u22122x\u22123<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2505172\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2505173\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2505172-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2933151\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\u00a0<\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>1<br \/>\nx<\/p>\n<p>=<br \/>\n1<br \/>\n5<\/p>\n<p>+<br \/>\n3<\/p>\n<p>2x<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2931117\">For the following exercises, find the equation of the line using the point-slope formula.<br \/>\nWrite all the final equations using the slope-intercept form.<\/p>\n<div id=\"fs-id2879070\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2879071\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>with a slope of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1956932\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1956933\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1956932-solution\">23<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>with a slope of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>4<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>4<\/mn><\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2020098\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2020099\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2020100\"><em data-effect=\"italics\">x<\/em>-intercept is 1, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1322919\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1322920\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1322919-solution\">25<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1388261\"><em data-effect=\"italics\">y<\/em>-intercept is 2, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2766380\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2683570\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>,<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>,<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2302767\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1687202\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2302767-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1760345\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2722754\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2722755\">parallel to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and passes through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1928538\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2698648\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1928538-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2698649\">perpendicular to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtext>3<\/mtext><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtext>3<\/mtext><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and passes through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1973853\">For the following exercises, find the equation of the line using the given information.<\/p>\n<div id=\"fs-id1712932\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1712933\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2433146\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2433147\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2433146-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>7<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2413598\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2413599\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2413600\">The slope is undefined and it passes through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1337290\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1337291\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1337290-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1337292\">The slope equals zero and it passes through the point<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>\u22124<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>\u22124<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3008660\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3008661\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3008662\">The\u00a0slope\u00a0is<\/p>\n<p><math display=\"inline\"><semantics><mrow><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac><\/annotation-xml><\/semantics><\/math>and\u00a0it\u00a0passes\u00a0through\u00a0the\u00a0point<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1545608\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1545609\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1545608-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>\u20131<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>\u20131<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>4<\/mn><mo>,<\/mo><mn>\u20135<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>4<\/mn><mo>,<\/mo><mn>\u20135<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id768131\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id2698028\">For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.<\/p>\n<div id=\"fs-id1719410\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1719412\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>1<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>1<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2515795\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2515797\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2515795-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2515798\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>6<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\u00a0<\/mtable><\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>3x\u22122y=5<\/p>\n<p>6y\u22129x=6<\/p>\n<p>\u00a0<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2736413\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1558245\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1558246\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\u00a0<\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/semantics><\/math><\/p>\n<p>y=3x+2<\/p>\n<p>y=<\/p>\n<p>3x+1<\/p>\n<p>4<br \/>\n\u00a0<\/p>\n<p>y=3x+2<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1942944\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1942945\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1942944-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1942946\"><math display=\"inline\"><semantics><mrow>\n<mtable columnalign=\"left\">\n<mtr>\n<mtd>\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mi>y<\/mi><mo>=<\/mo><mn>\u22123<\/mn>\n<\/mtd>\n<\/mtr>\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\n<p>x=4<\/p>\n<p>y=\u22123<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1700245\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1736475\">For the following exercises, find the slope of the line that passes through the given points.<\/p>\n<div id=\"fs-id1736478\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1736479\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>7<\/mn><mo>,<\/mo><mn>9<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>7<\/mn><mo>,<\/mo><mn>9<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1773932\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1895275\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1773932-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>\u22127<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mn>\u22127<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1846635\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1846636\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3160370\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3160371\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3160370-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2416770\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2416771\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo><mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1386040\">For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.<\/p>\n<div id=\"fs-id3165030\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3165031\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3165030-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3165032\"><math display=\"inline\"><semantics><mrow>\n<mtable columnalign=\"left\">\n<mtr>\n<mtd>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo><mn>9<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\n<p>(<\/p>\n<p>\u22121,3<\/p>\n<p>)\u00a0and\u00a0\u00a0(<\/p>\n<p>5,1<\/p>\n<p>)<\/p>\n<p>(<\/p>\n<p>\u22122,3<\/p>\n<p>)\u00a0and\u00a0\u00a0(<\/p>\n<p>0,9<\/p>\n<p>)<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2381892\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2381893\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2381894\"><math display=\"inline\"><semantics><mrow>\n<mtable columnalign=\"left\">\n<mtr>\n<mtd>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>,<\/mo><mn>9<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>\u00a0and\u00a0\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>,<\/mo><mn>3<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable>\u00a0<\/mrow><\/semantics><\/math><\/p>\n<p>(<\/p>\n<p>2,5<\/p>\n<p>)\u00a0and\u00a0\u00a0(<\/p>\n<p>5,9<\/p>\n<p>)<\/p>\n<p>(<\/p>\n<p>\u22121,\u22121<\/p>\n<p>)\u00a0and\u00a0\u00a0(<\/p>\n<p>2,3<\/p>\n<p>)<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2431191\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id2699691\">For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the <em data-effect=\"italics\">y<\/em>-intercept occurs. State your ymin and ymax values.<\/p>\n<div id=\"fs-id1390855\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1390856\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1390855-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0.537<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.19<\/mn><mi>y<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0.537<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.19<\/mn><mi>y<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2006746\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2006747\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4,500<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>200<\/mn><mi>y<\/mi><mo>=<\/mo><mn>9,528<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4,500<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>200<\/mn><mi>y<\/mi><mo>=<\/mo><mn>9,528<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2653806\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2653807\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2653806-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>200<\/mn><mo>\u2212<\/mo><mn>30<\/mn><mi>y<\/mi>\n<\/mrow>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mn>70<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>200<\/mn><mo>\u2212<\/mo><mn>30<\/mn><mi>y<\/mi>\n<\/mrow>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mn>70<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2294054\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id2387795\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2387796\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2387797\">Starting with the point-slope formula<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mi>m<\/mi><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>)<\/mo><mo>,<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mi>m<\/mi><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>)<\/mo><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>solve this expression for<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>x<\/mi><\/annotation-xml><\/semantics><\/math>in terms of<\/p>\n<p><math display=\"inline\"><semantics><mrow><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>y<\/mi><mo>,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>y<\/mi><mo>,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>m<\/mi><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2683690\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2683691\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2683690-solution\">51<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2683692\">Starting with the standard form of an equation<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>A<\/mi><mi>x<\/mi><mo>+<\/mo><mi>B<\/mi><mi>y<\/mi><mo>=<\/mo><mi>C<\/mi><\/annotation-xml><\/semantics><\/math>solve this expression for<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><\/annotation-xml><\/semantics><\/math>in terms of<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>A<\/mi><mo>,<\/mo><mi>B<\/mi><mo>,<\/mo><mi>C<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>A<\/mi><mo>,<\/mo><mi>B<\/mi><mo>,<\/mo><mi>C<\/mi><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>x<\/mi><\/annotation-xml><\/semantics><\/math>. Then put the expression in slope-intercept form.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2638987\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2638988\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2638989\">Use the above derived formula to put the following standard equation in slope intercept form:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>=<\/mo><mn>25.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>=<\/mo><mn>25.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2519655\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2519656\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2519655-solution\">53<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2519657\">Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>(<\/mo><mo>\u2013<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mo>\u2013<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>2<\/mn><mo>,<\/mo><mn>0<\/mn><mo>)<\/mo><mo>,<\/mo><mo>(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>3<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math><\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>(<\/mo><mn>0<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>(<\/mo><mn>0<\/mn><mo>,<\/mo><mn>4<\/mn><mo>)<\/mo><\/annotation-xml><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1972709\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1972710\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1972711\">Find the slopes of the diagonals in the previous exercise. Are they perpendicular?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1570540\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1274216\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1274217\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1274216-solution\">55<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1274218\">The slope for a wheelchair ramp for a home has to be<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.<\/p>\n<p><span id=\"fs-id1471955\" data-type=\"media\" data-alt=\"\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"\" width=\"402\" height=\"86\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/bded289bae0a02e51a163adfd8db52d5c7df5cc9\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3040342\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3040343\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3040344\">If the profit equation for a small business selling<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>number of item one and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>number of item two is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>p<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>p<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>find the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>value when<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>p<\/mi><mo>=<\/mo><mtext>$<\/mtext><mn>453<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0\u00a0<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>75.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>p<\/mi><mo>=<\/mo><mtext>$<\/mtext><mn>453<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0\u00a0<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>75.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id3231640\">For the following exercises, use this scenario: The cost of renting a car is $45\/wk plus $0.25\/mi traveled during that week. An equation to represent the cost would be<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>45<\/mn><mo>+<\/mo><mn>.25<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>45<\/mn><mo>+<\/mo><mn>.25<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>is the number of miles traveled.<\/p>\n<div id=\"fs-id3148386\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3148387\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3148386-solution\">57<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3148388\">What is your cost if you travel 50 mi?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1273127\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1273128\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1273129\">If your cost were<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtext>$<\/mtext><mn>63.75<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtext>$<\/mtext><mn>63.75<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>how many miles were you charged for traveling?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1315527\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1315528\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1315527-solution\">59<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1315529\">Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-108","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":3,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/108\/revisions"}],"predecessor-version":[{"id":319,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/108\/revisions\/319"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/108\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=108"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=108"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=108"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}