{"id":214,"date":"2025-04-09T17:28:54","date_gmt":"2025-04-09T17:28:54","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-3-systems-of-nonlinear-equations-and-inequalities-two-variables-college-algebra-2e-openstax\/"},"modified":"2025-09-03T19:14:03","modified_gmt":"2025-09-03T19:14:03","slug":"7-3-systems-of-nonlinear-equations-and-inequalities-two-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-3-systems-of-nonlinear-equations-and-inequalities-two-variables\/","title":{"raw":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","rendered":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_228bf1c6-e200-41b3-9bc0-79b6f6edbb37\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Solve a system of nonlinear equations using substitution.<\/li>\r\n \t<li>Solve a system of nonlinear equations using elimination.<\/li>\r\n \t<li>Graph a nonlinear inequality.<\/li>\r\n \t<li>Graph a system of nonlinear inequalities.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135558437\">Halley\u2019s Comet (Figure 1) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.<\/p>\r\n\r\n\r\n[caption id=\"attachment_1051\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1051\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-300x220.webp\" alt=\"\" width=\"300\" height=\"220\" \/> Figure 1. Halley\u2019s Comet (credit: \"NASA Blueshift\"\/Flickr)[\/caption]\r\n<p id=\"fs-id1165134189118\">In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.<\/p>\r\n\r\n<section id=\"fs-id1165134129918\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\r\n<p id=\"fs-id1165137783125\">A <strong>system of nonlinear equations<\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form [latex] Ax+By+c=0. [\/latex] Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.<\/p>\r\n\r\n<section id=\"fs-id1165135428383\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Intersection of a Parabola and a Line<\/h3>\r\n<p id=\"fs-id1165135367535\">There are three possible types of solutions for a system of nonlinear equations involving a <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span> and a line.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for Points of Intersection of a Parabola and a Line<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFigure 2 illustrates possible solution sets for a system of equations involving a parabola and a line.\r\n<ul>\r\n \t<li>No solution. The line will never intersect the parabola.<\/li>\r\n \t<li>One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.<\/li>\r\n \t<li>Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1053\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1053\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-300x103.webp\" alt=\"\" width=\"300\" height=\"103\" \/> Figure 2[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a system of equations containing a line and a parabola, find the solution.<\/strong>\r\n<ol>\r\n \t<li>Solve the linear equation for one of the variables.<\/li>\r\n \t<li>Substitute the expression obtained in step one into the parabola equation.<\/li>\r\n \t<li>Solve for the remaining variable.<\/li>\r\n \t<li>Check your solutions in both equations.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Solving a System of Nonlinear Equations Representing a Parabola and a Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-y &amp;=&amp; -1 \\\\ y &amp;=&amp; x^2+1 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Solve the first equation for <em>x<\/em> and then substitute the resulting expression into the second equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-y &amp;=&amp; -1 \\\\ x &amp;=&amp; y-1 &amp;&amp; \\text{Solve for } x. \\\\ \\\\ y &amp;=&amp; x^2+1 \\\\ y &amp;=&amp; (y-1)^2+1 &amp;&amp; \\text{Substitute expression for } x. \\end{array} [\/latex]<\/p>\r\nExpand the equation and set it equal to zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; (y-1)^2+1 \\\\ &amp;=&amp; (y^2-2y+1)+1 \\\\ &amp;=&amp; y^2-3y+2 \\\\ 0 &amp;=&amp; y^2-3y+2 \\\\ &amp;=&amp; (y-2)(y-1) \\end{array} [\/latex]<\/p>\r\nSolving for y gives [latex] y=2 [\/latex] and [latex] y=1. [\/latex] Next, substitute each value for <em>y<\/em> into the first equation to solve for <em>x<\/em>. Always substitute the value into the linear equation to check for extraneous solutions.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-y &amp;=&amp; -1 \\\\ x-(2) &amp;=&amp; -1 \\\\ x &amp;=&amp; 1 \\\\ \\\\ x-(1) &amp;=&amp; -1 \\\\ x &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\nThe solutions are [latex] (1, 2) [\/latex] and [latex] (0, 1) [\/latex] which can be verified by substituting these [latex] (x, y) [\/latex] values into both of the original equations. See Figure 3.\r\n\r\n[caption id=\"attachment_1054\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-1054\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 3[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Could we have substituted values for <em>y<\/em> into the second equation to solve for <em>x<\/em> in Example 1?<\/strong>\r\n\r\n<em>Yes, but because x is squared in the second equation this could give us extraneous solutions for x<\/em>\r\n\r\n<em>For [latex] y=1 [\/latex]<\/em>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; x^2+1 \\\\ 1 &amp;=&amp; x^2+1 \\\\ x^2 &amp;=&amp; 0 \\\\ x &amp;=&amp; \\pm\\sqrt{0}=0 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165134385530\"><em data-effect=\"italics\">This gives us the same value as in the solution.<\/em><\/p>\r\n<p id=\"fs-id1165137715303\"><em>For\u00a0 [latex] y=2 [\/latex]<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; x^2+1 \\\\ 2 &amp;=&amp; x^2+1 \\\\ x^2 &amp;=&amp; 1 \\\\ x &amp;=&amp; \\pm\\sqrt{1}=\\pm1 \\end{array} [\/latex]<\/p>\r\n<em>Notice that [latex] -1 [\/latex] is an extraneous solution.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the given system of equations by substitution.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3x-y &amp;=&amp; -2 \\\\ 2x^2-y &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135359807\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Intersection of a Circle and a Line<\/h3>\r\n<p id=\"fs-id1165135648585\">Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for the Points of Intersection of a Circle and a Line<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFigure 4 illustrates possible solution sets for a system of equations involving a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">circle<\/span> and a line.\r\n<ul>\r\n \t<li>No solution. The line does not intersect the circle.<\/li>\r\n \t<li>One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\r\n \t<li>Two solutions. The line crosses the circle and intersects it at two points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1056\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1056\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-300x109.webp\" alt=\"\" width=\"300\" height=\"109\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a system of equations containing a line and a circle, find the solution.<\/strong>\r\n<ol>\r\n \t<li>Solve the linear equation for one of the variables.<\/li>\r\n \t<li>Substitute the expression obtained in step one into the equation for the circle.<\/li>\r\n \t<li>Solve for the remaining variable.<\/li>\r\n \t<li>Check your solutions in both equations.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Finding the Intersection of a Circle and a Line by Substitution<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165135541669\">Find the intersection of the given circle and the given line by substitution.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 5 \\\\ y &amp;=&amp; 3x-5 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>One of the equations has already been solved for <em>y<\/em>. We will substitute [latex] y=3x-5 [\/latex] into the equation for the circle.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2+(3x-5)^2 &amp;=&amp; 5 \\\\ x^2+9x^2-30x+25 &amp;=&amp; 5 \\\\ 10x^2-30x+20 &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\nNow, we factor and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 10(x^2-3x+2) &amp;=&amp; 0 \\\\ 10(x-2)(x-1) &amp;=&amp; 0 \\\\ x &amp;=&amp; 2 \\\\ x &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\nSubstitute the two <em data-effect=\"italics\">x<\/em>-values into the original linear equation to solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; 3(2)-5 \\\\ &amp;=&amp; 1 \\\\ \\\\ y &amp;=&amp; 3(1)-5 \\\\ &amp;=&amp; -2 \\end{array} [\/latex]<\/p>\r\nThe line intersects the circle at [latex] (2, 1) [\/latex] and [latex]- (1, -2) [\/latex] which can be verified by substituting these [latex] (x, y) [\/latex] values into both of the original equations. See Figure 5.\r\n\r\n[caption id=\"attachment_1057\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1057\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-300x300.webp\" alt=\"\" width=\"300\" height=\"300\" \/> Figure 5[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">Try It #2<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165134325028\">Solve the system of nonlinear equations.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 10 \\\\ x-3y &amp;=&amp; -10 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135356442\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Elimination<\/h2>\r\n<p id=\"fs-id1165137628927\">We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">elimination<\/span> is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">circle<\/span> and an ellipse.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165132938353\">Figure 6 illustrates possible solution sets for a system of equations involving a circle and an <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">ellipse<\/span>.<\/p>\r\n\r\n<ul id=\"fs-id1165137694168\">\r\n \t<li>No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.<\/li>\r\n \t<li>One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\r\n \t<li>Two solutions. The circle and the ellipse intersect at two points.<\/li>\r\n \t<li>Three solutions. The circle and the ellipse intersect at three points.<\/li>\r\n \t<li>Four solutions. The circle and the ellipse intersect at four points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1058\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1058\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-300x74.webp\" alt=\"\" width=\"300\" height=\"74\" \/> Figure 6[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of nonlinear equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 26 &amp;&amp; (1) \\\\ 3x^2+25y^2 &amp;=&amp; 100 &amp;&amp; (2) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Let\u2019s begin by multiplying equation (1) by [latex] -3, [\/latex] and adding it to equation (2).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} (-3)(x^2+y^2) &amp;=&amp; (-3)(26) \\\\ -3x^2-3y^2 &amp;=&amp; -78 \\\\ 3x^2+25y^2 &amp;=&amp; 100 \\\\ \\hline 22y^2 &amp;=&amp; 22 \\end{array} [\/latex]<\/p>\r\nAfter we add the two equations together, we solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y^2 &amp;=&amp; 1 \\\\ y &amp;=&amp; \\pm\\sqrt{1}=\\pm1 \\end{array} [\/latex]<\/p>\r\nSubstitute [latex] y=\\pm1 [\/latex] into one of the equations and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2+(1)^2 &amp;=&amp; 26 \\\\ x^2+1 &amp;=&amp; 26 \\\\ x^2 &amp;=&amp; 25 \\\\ x &amp;=&amp; \\pm\\sqrt{25}=\\pm5 \\\\ \\\\ x^2+(-1)^2 &amp;=&amp; 26 \\\\ x^2+1 &amp;=&amp; 26 \\\\ x^2 &amp;=&amp; 25=\\pm5 \\end{array} [\/latex]<\/p>\r\nThere are four solutions: [latex] (5, 1), (-5, 1) (5, -1) [\/latex] and [latex] (-5, -1). [\/latex] See Figure 7.\r\n\r\n[caption id=\"attachment_1059\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1059\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-300x259.webp\" alt=\"\" width=\"300\" height=\"259\" \/> Figure 7[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165135192764\">Find the solution set for the given system of nonlinear equations.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 4x^2+y^2 &amp;=&amp; 13 \\\\ x^2+y^2 &amp;=&amp; 10 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165134361408\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing a Nonlinear Inequality<\/h2>\r\n<p id=\"fs-id1165137657083\">All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A <strong>nonlinear inequality<\/strong> is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.<\/p>\r\n<p id=\"fs-id1165135295026\">Recall that when the inequality is greater than, [latex] y&gt; 1, [\/latex] or less than, [latex] y&lt; a, [\/latex] the graph is drawn with a dashed line. When the inequality is greater than or equal to, [latex] y\\ge 1, [\/latex] or less than or equal to, [latex] y\\le a, [\/latex] the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See Figure 8.<\/p>\r\n\r\n[caption id=\"attachment_1060\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1060\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-300x144.webp\" alt=\"\" width=\"300\" height=\"144\" \/> Figure 8. (a) an example of [latex] y&gt; a; [\/latex] (b) an example of [latex] y\\ge a; [\/latex] (c) an example of [latex] y&lt; a; [\/latex] (d) an example of [latex] y\\le a. [\/latex][\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165137871377\"><strong>Given an inequality bounded by a parabola, sketch a graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137896994\" type=\"1\">\r\n \t<li>Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.<\/li>\r\n \t<li>If the boundary is included in the region (the operator is [latex] \\le [\/latex] or [latex] \\ge [\/latex]), the parabola is graphed as a solid line.<\/li>\r\n \t<li>If the boundary is not included in the region (the operator is [latex] &lt; [\/latex] or [latex] &gt; [\/latex]), the parabola is graphed as a dashed line.<\/li>\r\n \t<li>Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is true, the solution set is the region including the point. If the statement is false, the solution set is the region on the other side of the boundary line.<\/li>\r\n \t<li>Shade the region representing the solution set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Graphing an Inequality for a Parabola<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph the inequality [latex] y&gt; x^2+1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, graph the corresponding equation [latex] y=x^2+1. [\/latex] Since [latex] y&gt; x^2+1 [\/latex] has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let\u2019s test the points [latex] (0, 2) [\/latex] and [latex] (2, 0). [\/latex] One point is clearly inside the parabola and the other point is clearly outside.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;&gt;&amp; x^2+1 \\\\ 2 &amp;&gt;&amp; (0)^2+1 \\\\ 2 &amp;&gt;&amp; 1 &amp;&amp; \\text{True} \\\\ \\\\ 0 &amp;&gt;&amp; (2)^2+1 \\\\ 0 &amp;&gt;&amp; 5 &amp;&amp; \\text{False} \\end{array} [\/latex]<\/p>\r\nThe graph is shown in Figure 9. We can see that the solution set consists of all points inside the parabola, but not on the graph itself.\r\n\r\n[caption id=\"attachment_1061\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1061\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-300x211.webp\" alt=\"\" width=\"300\" height=\"211\" \/> Figure 9[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165133050485\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing a System of Nonlinear Inequalities<\/h2>\r\n<p id=\"fs-id1165137855322\">Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A <strong>system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the <strong>feasible region<\/strong>.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165137835837\"><strong>Given a system of nonlinear inequalities, sketch a graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137465479\" type=\"1\">\r\n \t<li>Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\r\n \t<li>Graph the nonlinear equations.<\/li>\r\n \t<li>Find the shaded regions of each inequality.<\/li>\r\n \t<li>Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Graphing a System of Inequalities<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph the given system of inequalities.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2-y &amp;\\le&amp; 0 \\\\ 2x^2+y &amp;\\le&amp; 12 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable y will be eliminated. Then we solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2-y &amp;=&amp; 0 \\\\ 2x^2+y &amp;=&amp; 12 \\\\ \\hline 3x^2 &amp;=&amp; 12 \\\\ x^2 &amp;=&amp; 4 \\\\ x &amp;=&amp; \\pm2 \\end{array} [\/latex]<\/p>\r\nSubstitute the <em data-effect=\"italics\">x<\/em>-values into one of the equations and solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2-y &amp;=&amp; 0\\\\ (2)^2-y &amp;=&amp; 0 \\\\ 4-y &amp;=&amp; 0 \\\\ y &amp;=&amp; 4 \\end{array} [\/latex]<\/p>\r\nThe two points of intersection are [latex] (2, 4) [\/latex] and [latex] (-2, 4). [\/latex] Notice that the equations can be rewritten as follows.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x^2-y &amp;\\le&amp; 0 \\\\ x^2 &amp;\\le&amp; y \\\\ y &amp;\\ge&amp; x^2 \\\\ \\\\ 2x^2+y &amp;\\le&amp; 12 \\\\ y &amp;\\le&amp; -2x^2+12 \\end{array} [\/latex]<\/p>\r\nGraph each inequality. See Figure 10. The feasible region is the region between the two equations bounded by [latex] 2x^2+y\\le 12 [\/latex] on the top and [latex] x^2-y\\le 0 [\/latex] on the bottom.\r\n\r\n[caption id=\"attachment_1062\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1062\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-300x235.webp\" alt=\"\" width=\"300\" height=\"235\" \/> Figure 10[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165137900185\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137768089\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165134357579\">Graph the given system of inequalities.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;\\ge&amp; x^2-1 \\\\ x-y &amp;\\ge&amp; -1 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with nonlinear equations.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=VI4V_X_ZZLc\">Solve a System of Nonlinear Equations Using Substitution<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=Ic-42kmdJqc\">Solve a System of Nonlinear Equations Using Elimination<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\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\">7.3 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137404664\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135664812\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137629528\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137629529\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137629528-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135509064\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135359763\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137437074\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137437075\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137437074-solution\">3<\/a><span class=\"os-divider\">. <\/span>When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137897804\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137897805\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If you graph a revenue and cost function, explain how to determine in what regions there is profit.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137771182\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137771184\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137771182-solution\">5<\/a><span class=\"os-divider\">. <\/span>If you perform your break-even analysis and there is more than one solution, explain how you would determine which <em data-effect=\"italics\">x<\/em>-values are profit and which are not.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137451079\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165134069312\">For the following exercises, solve the system of nonlinear equations using substitution.<\/p>\r\n\r\n<div id=\"fs-id1165134069315\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134069316\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y &amp;=&amp; 4 \\\\ x^2+y^2 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165135199365\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135199366\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135199365-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y &amp;=&amp; x-3 \\\\ x^2+y^2 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137695996\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137695997\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y &amp;=&amp; x \\\\ x^2+y^2 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135415865\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135415866\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135415865-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y &amp;=&amp; -x \\\\ x^2+y^2 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134298447\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134298448\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x &amp;=&amp; 2 \\\\ x^2-y^2 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137462986\">For the following exercises, solve the system of nonlinear equations using elimination.<\/p>\r\n\r\n<div id=\"fs-id1165137462989\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137462990\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137462989-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 4x^2-9y^2 &amp;=&amp; 36 \\\\ 4x^2+9y^2 &amp;=&amp; 36 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137663022\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137663023\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 25 \\\\ x^2-y^2 &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135481002\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135481003\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135481002-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x^2+4y^2 &amp;=&amp; 4 \\\\ 2x^2-4y^2 &amp;=&amp; 25x-10 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135342197\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135342198\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y^2-x^2 &amp;=&amp; 9 \\\\ 3x^2+2y^2 &amp;=&amp; 8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135400179\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137602296\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135400179-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2+\\frac{1}{16} &amp;=&amp; 2500 \\\\ y &amp;=&amp; 2x^2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165133075641\">For the following exercises, use any method to solve the system of nonlinear equations.<\/p>\r\n\r\n<div id=\"fs-id1165134401678\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134401679\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -2x^2+y &amp;=&amp; -5 \\\\ 6x-y &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135518200\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135518201\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135518200-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x^2+y &amp;=&amp; 2 \\\\ -x+y &amp;=&amp; 2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133409834\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133409835\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 1 \\\\ y &amp;=&amp; 20x^2-1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133074924\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135300718\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133074924-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 1 \\\\ y &amp;=&amp; -x^2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134418897\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134418898\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x^3-x^2 &amp;=&amp; y \\\\ y &amp;=&amp; \\frac{1}{2}-x \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134346292\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134346293\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134346292-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 9x^2+25y^2 &amp;=&amp; 225 \\\\ (x-6)^2+y^2 &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132959043\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132959044\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^4-x^2 &amp;=&amp; y \\\\ x^2+y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134054828\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134054830\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134054828-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x^3-x^2 &amp;=&amp; y \\\\ x^2+y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135182826\">For the following exercises, use any method to solve the nonlinear system.<\/p>\r\n\r\n<div id=\"fs-id1165135182829\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135182830\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 9 \\\\ y &amp;=&amp; 3-x^2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133026273\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133026274\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133026273-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-y^2 &amp;=&amp; 9 \\\\ x &amp;=&amp; 3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134180366\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134180367\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-y^2 &amp;=&amp; 9 \\\\ y &amp;=&amp; 3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133308372\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133308373\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133308372-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-y^2 &amp;=&amp; 9 \\\\ x-y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134252122\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134252123\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x^2+y &amp;=&amp; 2 \\\\ -4x+y &amp;=&amp; -1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133015726\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133015728\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133015726-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x^2+y &amp;=&amp; 2 \\\\ 2y &amp;=&amp; -x \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133155744\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133155745\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 25 \\\\ x^2-y^2 &amp;=&amp; 36 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131995098\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131995099\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131995098-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 1 \\\\ y^2 &amp;=&amp; x^2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134367925\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134367926\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 16x^2-9y^2+144 &amp;=&amp; 0 \\\\ y^2+x^2 &amp;=&amp; 16 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135330972\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135330974\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135330972-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x^2-y^2 &amp;=&amp; 12 \\\\ (x-1)^2+y^2 &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133201780\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133201781\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x^2-y^2 &amp;=&amp; 12 \\\\ (x-1)^2+y^2 &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135425726\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135425727\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135425726-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x^2-y^2 &amp;=&amp; 12 \\\\ x^2+y^2 &amp;=&amp; 16 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137394072\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137394073\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-y^2-6x-4y-11 &amp;=&amp; 0 \\\\ -x^2+y^2 &amp;=&amp; 5 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134058210\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134058211\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134058210-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2-6y &amp;=&amp; 7 \\\\ x^2+y &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134231179\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134231180\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;=&amp; 6 \\\\ xy &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135451239\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135451244\">For the following exercises, graph the inequality.<\/p>\r\n\r\n<div id=\"fs-id1165135451248\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135451249\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135451248-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] x^2+y&lt; 9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165131924514\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131924515\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex0 x^2+y^2&lt; 4] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135669591\">For the following exercises, graph the system of inequalities. Label all points of intersection.<\/p>\r\n\r\n<div id=\"fs-id1165135669594\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135176855\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135669594-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y &amp;&lt;&amp; 1 \\\\ y &amp;&gt;&amp; 2x \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134544651\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134544652\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y &amp;&lt;&amp; -5 \\\\ y &amp;&gt;&amp; 5x+10 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133404494\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133404495\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133404494-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y^2 &amp;&lt;&amp; 25 \\\\ 3x^2-y^2 &amp;&gt;&amp; 12 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135491494\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135491495\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-y^2 &amp;&gt;&amp; -4 \\\\ x^2+y^2 &amp;&lt;&amp; 12 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135474650\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135474651\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135474650-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+3y^2 &amp;&gt;&amp; 16 \\\\ 3x^2-y^2 &amp;&lt;&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135680032\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165135680037\">For the following exercises, graph the inequality.<\/p>\r\n\r\n<div id=\"fs-id1165135680040\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135680041\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y &amp;\\ge&amp; e^x \\\\ y &amp;\\le&amp; \\ln(x)+5 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165133183427\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133183428\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133183427-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} y &amp;\\le&amp; -\\log(x) \\\\ y &amp;\\le&amp; e^x \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134185314\">For the following exercises, find the solutions to the nonlinear equations with two variables.<\/p>\r\n\r\n<div id=\"fs-id1165134185317\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134185318\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{4}{x^2}+\\frac{1}{y^2} &amp;=&amp; 24 \\\\ \\frac{5}{x^2}-\\frac{2}{y^2}+4 &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131911076\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131911077\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131911076-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{6}{x^2}-\\frac{1}{y^2} &amp;=&amp; 8 \\\\ \\frac{1}{x^2}-\\frac{6}{y^2} &amp;=&amp; \\frac{1}{8} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135461999\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135462000\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-xy+y^2-2 &amp;=&amp; 0 \\\\ x+3y &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134223178\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134223179\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134223178-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2-xy-2y^2-6 &amp;=&amp; 0 \\\\ x^2+y^2 &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132213505\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132213506\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+4xy-2y^2-6 &amp;=&amp; 0 \\\\ x &amp;=&amp; y+2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134237067\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165134237072\">For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.<\/p>\r\n\r\n<div id=\"fs-id1165135239654\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135239655\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135239654-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} xy &amp;&lt;&amp; 1 \\\\ y &amp;&gt;&amp; \\sqrt{x} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165134181692\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134181693\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x^2+y &amp;&lt;&amp; 3 \\\\ y &amp;&gt;&amp; 2x \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134183638\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1165133007623\">For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.<\/p>\r\n\r\n<div id=\"fs-id1165133007628\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133007629\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133007628-solution\">55<\/a><span class=\"os-divider\">. <\/span>Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135602212\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135602213\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>The squares of two numbers add to 360. The second number is half the value of the first number squared. What are the numbers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135602219\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135602220\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135602219-solution\">57<\/a><span class=\"os-divider\">. <\/span>A laptop company has discovered their cost and revenue functions for each day: [latex] C(x)=3x^2-10x+200 [\/latex] and [latex] R(x)=-2x^2+100x+50. [\/latex] If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135695987\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135695988\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>A cell phone company has the following cost and revenue functions: [latex] C(x)=8x^2-600x+21,500 [\/latex] and [latex] R(x)=-3x^2+480x. [\/latex] What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_228bf1c6-e200-41b3-9bc0-79b6f6edbb37\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Solve a system of nonlinear equations using substitution.<\/li>\n<li>Solve a system of nonlinear equations using elimination.<\/li>\n<li>Graph a nonlinear inequality.<\/li>\n<li>Graph a system of nonlinear inequalities.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135558437\">Halley\u2019s Comet (Figure 1) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.<\/p>\n<figure id=\"attachment_1051\" aria-describedby=\"caption-attachment-1051\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1051\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-300x220.webp\" alt=\"\" width=\"300\" height=\"220\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-300x220.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-65x48.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-225x165.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1-350x257.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-1.webp 477w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1051\" class=\"wp-caption-text\">Figure 1. Halley\u2019s Comet (credit: &#8220;NASA Blueshift&#8221;\/Flickr)<\/figcaption><\/figure>\n<p id=\"fs-id1165134189118\">In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.<\/p>\n<section id=\"fs-id1165134129918\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\n<p id=\"fs-id1165137783125\">A <strong>system of nonlinear equations<\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form [latex]Ax+By+c=0.[\/latex] Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.<\/p>\n<section id=\"fs-id1165135428383\" data-depth=\"2\">\n<h3 data-type=\"title\">Intersection of a Parabola and a Line<\/h3>\n<p id=\"fs-id1165135367535\">There are three possible types of solutions for a system of nonlinear equations involving a <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span> and a line.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for Points of Intersection of a Parabola and a Line<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Figure 2 illustrates possible solution sets for a system of equations involving a parabola and a line.<\/p>\n<ul>\n<li>No solution. The line will never intersect the parabola.<\/li>\n<li>One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.<\/li>\n<li>Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.<\/li>\n<\/ul>\n<figure id=\"attachment_1053\" aria-describedby=\"caption-attachment-1053\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1053\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-300x103.webp\" alt=\"\" width=\"300\" height=\"103\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-300x103.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-768x264.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-65x22.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-225x77.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1-350x120.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-2-1.webp 944w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1053\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a system of equations containing a line and a parabola, find the solution.<\/strong><\/p>\n<ol>\n<li>Solve the linear equation for one of the variables.<\/li>\n<li>Substitute the expression obtained in step one into the parabola equation.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Check your solutions in both equations.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Solving a System of Nonlinear Equations Representing a Parabola and a Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-y &=& -1 \\\\ y &=& x^2+1 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Solve the first equation for <em>x<\/em> and then substitute the resulting expression into the second equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-y &=& -1 \\\\ x &=& y-1 && \\text{Solve for } x. \\\\ \\\\ y &=& x^2+1 \\\\ y &=& (y-1)^2+1 && \\text{Substitute expression for } x. \\end{array}[\/latex]<\/p>\n<p>Expand the equation and set it equal to zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& (y-1)^2+1 \\\\ &=& (y^2-2y+1)+1 \\\\ &=& y^2-3y+2 \\\\ 0 &=& y^2-3y+2 \\\\ &=& (y-2)(y-1) \\end{array}[\/latex]<\/p>\n<p>Solving for y gives [latex]y=2[\/latex] and [latex]y=1.[\/latex] Next, substitute each value for <em>y<\/em> into the first equation to solve for <em>x<\/em>. Always substitute the value into the linear equation to check for extraneous solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-y &=& -1 \\\\ x-(2) &=& -1 \\\\ x &=& 1 \\\\ \\\\ x-(1) &=& -1 \\\\ x &=& 0 \\end{array}[\/latex]<\/p>\n<p>The solutions are [latex](1, 2)[\/latex] and [latex](0, 1)[\/latex] which can be verified by substituting these [latex](x, y)[\/latex] values into both of the original equations. See Figure 3.<\/p>\n<figure id=\"attachment_1054\" aria-describedby=\"caption-attachment-1054\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1054\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-3.webp 384w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-1054\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Could we have substituted values for <em>y<\/em> into the second equation to solve for <em>x<\/em> in Example 1?<\/strong><\/p>\n<p><em>Yes, but because x is squared in the second equation this could give us extraneous solutions for x<\/em><\/p>\n<p><em>For [latex]y=1[\/latex]<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& x^2+1 \\\\ 1 &=& x^2+1 \\\\ x^2 &=& 0 \\\\ x &=& \\pm\\sqrt{0}=0 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165134385530\"><em data-effect=\"italics\">This gives us the same value as in the solution.<\/em><\/p>\n<p id=\"fs-id1165137715303\"><em>For\u00a0 [latex]y=2[\/latex]<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& x^2+1 \\\\ 2 &=& x^2+1 \\\\ x^2 &=& 1 \\\\ x &=& \\pm\\sqrt{1}=\\pm1 \\end{array}[\/latex]<\/p>\n<p><em>Notice that [latex]-1[\/latex] is an extraneous solution.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the given system of equations by substitution.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3x-y &=& -2 \\\\ 2x^2-y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135359807\" data-depth=\"2\">\n<h3 data-type=\"title\">Intersection of a Circle and a Line<\/h3>\n<p id=\"fs-id1165135648585\">Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for the Points of Intersection of a Circle and a Line<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Figure 4 illustrates possible solution sets for a system of equations involving a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">circle<\/span> and a line.<\/p>\n<ul>\n<li>No solution. The line does not intersect the circle.<\/li>\n<li>One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\n<li>Two solutions. The line crosses the circle and intersects it at two points.<\/li>\n<\/ul>\n<figure id=\"attachment_1056\" aria-describedby=\"caption-attachment-1056\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1056\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-300x109.webp\" alt=\"\" width=\"300\" height=\"109\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-300x109.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-65x24.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-225x82.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4-350x127.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-4.webp 717w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1056\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a system of equations containing a line and a circle, find the solution.<\/strong><\/p>\n<ol>\n<li>Solve the linear equation for one of the variables.<\/li>\n<li>Substitute the expression obtained in step one into the equation for the circle.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Check your solutions in both equations.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Finding the Intersection of a Circle and a Line by Substitution<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165135541669\">Find the intersection of the given circle and the given line by substitution.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2+y^2 &=& 5 \\\\ y &=& 3x-5 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>One of the equations has already been solved for <em>y<\/em>. We will substitute [latex]y=3x-5[\/latex] into the equation for the circle.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2+(3x-5)^2 &=& 5 \\\\ x^2+9x^2-30x+25 &=& 5 \\\\ 10x^2-30x+20 &=& 0 \\end{array}[\/latex]<\/p>\n<p>Now, we factor and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 10(x^2-3x+2) &=& 0 \\\\ 10(x-2)(x-1) &=& 0 \\\\ x &=& 2 \\\\ x &=& 1 \\end{array}[\/latex]<\/p>\n<p>Substitute the two <em data-effect=\"italics\">x<\/em>-values into the original linear equation to solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& 3(2)-5 \\\\ &=& 1 \\\\ \\\\ y &=& 3(1)-5 \\\\ &=& -2 \\end{array}[\/latex]<\/p>\n<p>The line intersects the circle at [latex](2, 1)[\/latex] and [latex]- (1, -2)[\/latex] which can be verified by substituting these [latex](x, y)[\/latex] values into both of the original equations. See Figure 5.<\/p>\n<figure id=\"attachment_1057\" aria-describedby=\"caption-attachment-1057\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1057\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-300x300.webp\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-300x300.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-65x65.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-225x226.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5-350x351.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-5.webp 363w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1057\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">Try It #2<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165134325028\">Solve the system of nonlinear equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2+y^2 &=& 10 \\\\ x-3y &=& -10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135356442\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Elimination<\/h2>\n<p id=\"fs-id1165137628927\">We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">elimination<\/span> is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">circle<\/span> and an ellipse.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165132938353\">Figure 6 illustrates possible solution sets for a system of equations involving a circle and an <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">ellipse<\/span>.<\/p>\n<ul id=\"fs-id1165137694168\">\n<li>No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.<\/li>\n<li>One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\n<li>Two solutions. The circle and the ellipse intersect at two points.<\/li>\n<li>Three solutions. The circle and the ellipse intersect at three points.<\/li>\n<li>Four solutions. The circle and the ellipse intersect at four points.<\/li>\n<\/ul>\n<figure id=\"attachment_1058\" aria-describedby=\"caption-attachment-1058\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1058\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-300x74.webp\" alt=\"\" width=\"300\" height=\"74\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-300x74.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-1024x254.webp 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-768x191.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-65x16.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-225x56.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6-350x87.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-6.webp 1108w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1058\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system of nonlinear equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2+y^2 &=& 26 && (1) \\\\ 3x^2+25y^2 &=& 100 && (2) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Let\u2019s begin by multiplying equation (1) by [latex]-3,[\/latex] and adding it to equation (2).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} (-3)(x^2+y^2) &=& (-3)(26) \\\\ -3x^2-3y^2 &=& -78 \\\\ 3x^2+25y^2 &=& 100 \\\\ \\hline 22y^2 &=& 22 \\end{array}[\/latex]<\/p>\n<p>After we add the two equations together, we solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y^2 &=& 1 \\\\ y &=& \\pm\\sqrt{1}=\\pm1 \\end{array}[\/latex]<\/p>\n<p>Substitute [latex]y=\\pm1[\/latex] into one of the equations and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2+(1)^2 &=& 26 \\\\ x^2+1 &=& 26 \\\\ x^2 &=& 25 \\\\ x &=& \\pm\\sqrt{25}=\\pm5 \\\\ \\\\ x^2+(-1)^2 &=& 26 \\\\ x^2+1 &=& 26 \\\\ x^2 &=& 25=\\pm5 \\end{array}[\/latex]<\/p>\n<p>There are four solutions: [latex](5, 1), (-5, 1) (5, -1)[\/latex] and [latex](-5, -1).[\/latex] See Figure 7.<\/p>\n<figure id=\"attachment_1059\" aria-describedby=\"caption-attachment-1059\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1059\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-300x259.webp\" alt=\"\" width=\"300\" height=\"259\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-300x259.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-65x56.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-225x194.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7-350x302.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-7.webp 429w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1059\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165135192764\">Find the solution set for the given system of nonlinear equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 4x^2+y^2 &=& 13 \\\\ x^2+y^2 &=& 10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134361408\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing a Nonlinear Inequality<\/h2>\n<p id=\"fs-id1165137657083\">All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A <strong>nonlinear inequality<\/strong> is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.<\/p>\n<p id=\"fs-id1165135295026\">Recall that when the inequality is greater than, [latex]y> 1,[\/latex] or less than, [latex]y< a,[\/latex] the graph is drawn with a dashed line. When the inequality is greater than or equal to, [latex]y\\ge 1,[\/latex] or less than or equal to, [latex]y\\le a,[\/latex] the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See Figure 8.<\/p>\n<figure id=\"attachment_1060\" aria-describedby=\"caption-attachment-1060\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1060\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-300x144.webp\" alt=\"\" width=\"300\" height=\"144\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-300x144.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-768x369.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-65x31.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-225x108.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8-350x168.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-8.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1060\" class=\"wp-caption-text\">Figure 8. (a) an example of [latex] y&gt; a; [\/latex] (b) an example of [latex] y\\ge a; [\/latex] (c) an example of [latex] y&lt; a; [\/latex] (d) an example of [latex] y\\le a. [\/latex]<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165137871377\"><strong>Given an inequality bounded by a parabola, sketch a graph.<\/strong><\/p>\n<ol id=\"fs-id1165137896994\" type=\"1\">\n<li>Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.<\/li>\n<li>If the boundary is included in the region (the operator is [latex]\\le[\/latex] or [latex]\\ge[\/latex]), the parabola is graphed as a solid line.<\/li>\n<li>If the boundary is not included in the region (the operator is [latex]<[\/latex] or [latex]>[\/latex]), the parabola is graphed as a dashed line.<\/li>\n<li>Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is true, the solution set is the region including the point. If the statement is false, the solution set is the region on the other side of the boundary line.<\/li>\n<li>Shade the region representing the solution set.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Graphing an Inequality for a Parabola<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the inequality [latex]y> x^2+1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, graph the corresponding equation [latex]y=x^2+1.[\/latex] Since [latex]y> x^2+1[\/latex] has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let\u2019s test the points [latex](0, 2)[\/latex] and [latex](2, 0).[\/latex] One point is clearly inside the parabola and the other point is clearly outside.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &>& x^2+1 \\\\ 2 &>& (0)^2+1 \\\\ 2 &>& 1 && \\text{True} \\\\ \\\\ 0 &>& (2)^2+1 \\\\ 0 &>& 5 && \\text{False} \\end{array}[\/latex]<\/p>\n<p>The graph is shown in Figure 9. We can see that the solution set consists of all points inside the parabola, but not on the graph itself.<\/p>\n<figure id=\"attachment_1061\" aria-describedby=\"caption-attachment-1061\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1061\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-300x211.webp\" alt=\"\" width=\"300\" height=\"211\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-300x211.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-65x46.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-225x158.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9-350x246.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-9.webp 353w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1061\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133050485\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing a System of Nonlinear Inequalities<\/h2>\n<p id=\"fs-id1165137855322\">Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A <strong>system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the <strong>feasible region<\/strong>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165137835837\"><strong>Given a system of nonlinear inequalities, sketch a graph.<\/strong><\/p>\n<ol id=\"fs-id1165137465479\" type=\"1\">\n<li>Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\n<li>Graph the nonlinear equations.<\/li>\n<li>Find the shaded regions of each inequality.<\/li>\n<li>Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Graphing a System of Inequalities<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the given system of inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2-y &\\le& 0 \\\\ 2x^2+y &\\le& 12 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable y will be eliminated. Then we solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2-y &=& 0 \\\\ 2x^2+y &=& 12 \\\\ \\hline 3x^2 &=& 12 \\\\ x^2 &=& 4 \\\\ x &=& \\pm2 \\end{array}[\/latex]<\/p>\n<p>Substitute the <em data-effect=\"italics\">x<\/em>-values into one of the equations and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2-y &=& 0\\\\ (2)^2-y &=& 0 \\\\ 4-y &=& 0 \\\\ y &=& 4 \\end{array}[\/latex]<\/p>\n<p>The two points of intersection are [latex](2, 4)[\/latex] and [latex](-2, 4).[\/latex] Notice that the equations can be rewritten as follows.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x^2-y &\\le& 0 \\\\ x^2 &\\le& y \\\\ y &\\ge& x^2 \\\\ \\\\ 2x^2+y &\\le& 12 \\\\ y &\\le& -2x^2+12 \\end{array}[\/latex]<\/p>\n<p>Graph each inequality. See Figure 10. The feasible region is the region between the two equations bounded by [latex]2x^2+y\\le 12[\/latex] on the top and [latex]x^2-y\\le 0[\/latex] on the bottom.<\/p>\n<figure id=\"attachment_1062\" aria-describedby=\"caption-attachment-1062\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1062\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-300x235.webp\" alt=\"\" width=\"300\" height=\"235\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-300x235.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-65x51.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-225x176.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10-350x274.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.3-fig-10.webp 353w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1062\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165137900185\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137768089\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165134357579\">Graph the given system of inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &\\ge& x^2-1 \\\\ x-y &\\ge& -1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with nonlinear equations.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=VI4V_X_ZZLc\">Solve a System of Nonlinear Equations Using Substitution<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=Ic-42kmdJqc\">Solve a System of Nonlinear Equations Using Elimination<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\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\">7.3 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137404664\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135664812\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137629528\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137629529\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137629528-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135509064\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135359763\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137437074\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137437075\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137437074-solution\">3<\/a><span class=\"os-divider\">. <\/span>When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137897804\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137897805\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If you graph a revenue and cost function, explain how to determine in what regions there is profit.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137771182\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137771184\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137771182-solution\">5<\/a><span class=\"os-divider\">. <\/span>If you perform your break-even analysis and there is more than one solution, explain how you would determine which <em data-effect=\"italics\">x<\/em>-values are profit and which are not.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137451079\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165134069312\">For the following exercises, solve the system of nonlinear equations using substitution.<\/p>\n<div id=\"fs-id1165134069315\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134069316\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y &=& 4 \\\\ x^2+y^2 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135199365\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135199366\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135199365-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y &=& x-3 \\\\ x^2+y^2 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137695996\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137695997\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y &=& x \\\\ x^2+y^2 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135415865\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135415866\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135415865-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y &=& -x \\\\ x^2+y^2 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134298447\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134298448\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x &=& 2 \\\\ x^2-y^2 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137462986\">For the following exercises, solve the system of nonlinear equations using elimination.<\/p>\n<div id=\"fs-id1165137462989\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137462990\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137462989-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 4x^2-9y^2 &=& 36 \\\\ 4x^2+9y^2 &=& 36 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137663022\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137663023\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 25 \\\\ x^2-y^2 &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135481002\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135481003\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135481002-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x^2+4y^2 &=& 4 \\\\ 2x^2-4y^2 &=& 25x-10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135342197\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135342198\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y^2-x^2 &=& 9 \\\\ 3x^2+2y^2 &=& 8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135400179\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137602296\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135400179-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2+\\frac{1}{16} &=& 2500 \\\\ y &=& 2x^2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165133075641\">For the following exercises, use any method to solve the system of nonlinear equations.<\/p>\n<div id=\"fs-id1165134401678\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134401679\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -2x^2+y &=& -5 \\\\ 6x-y &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135518200\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135518201\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135518200-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x^2+y &=& 2 \\\\ -x+y &=& 2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133409834\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133409835\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 1 \\\\ y &=& 20x^2-1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133074924\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135300718\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133074924-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 1 \\\\ y &=& -x^2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134418897\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134418898\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x^3-x^2 &=& y \\\\ y &=& \\frac{1}{2}-x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134346292\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134346293\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134346292-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 9x^2+25y^2 &=& 225 \\\\ (x-6)^2+y^2 &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132959043\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132959044\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^4-x^2 &=& y \\\\ x^2+y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134054828\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134054830\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134054828-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x^3-x^2 &=& y \\\\ x^2+y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135182826\">For the following exercises, use any method to solve the nonlinear system.<\/p>\n<div id=\"fs-id1165135182829\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135182830\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 9 \\\\ y &=& 3-x^2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133026273\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133026274\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133026273-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-y^2 &=& 9 \\\\ x &=& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134180366\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134180367\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-y^2 &=& 9 \\\\ y &=& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133308372\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133308373\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133308372-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-y^2 &=& 9 \\\\ x-y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134252122\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134252123\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x^2+y &=& 2 \\\\ -4x+y &=& -1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133015726\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133015728\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133015726-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x^2+y &=& 2 \\\\ 2y &=& -x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133155744\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133155745\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 25 \\\\ x^2-y^2 &=& 36 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131995098\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131995099\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131995098-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 1 \\\\ y^2 &=& x^2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134367925\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134367926\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 16x^2-9y^2+144 &=& 0 \\\\ y^2+x^2 &=& 16 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135330972\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135330974\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135330972-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x^2-y^2 &=& 12 \\\\ (x-1)^2+y^2 &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133201780\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133201781\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x^2-y^2 &=& 12 \\\\ (x-1)^2+y^2 &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135425726\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135425727\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135425726-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x^2-y^2 &=& 12 \\\\ x^2+y^2 &=& 16 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137394072\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137394073\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-y^2-6x-4y-11 &=& 0 \\\\ -x^2+y^2 &=& 5 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134058210\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134058211\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134058210-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2-6y &=& 7 \\\\ x^2+y &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134231179\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134231180\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &=& 6 \\\\ xy &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135451239\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135451244\">For the following exercises, graph the inequality.<\/p>\n<div id=\"fs-id1165135451248\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135451249\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135451248-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]x^2+y< 9[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165131924514\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131924515\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex0 x^2+y^2&lt; 4] [\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135669591\">For the following exercises, graph the system of inequalities. Label all points of intersection.<\/p>\n<div id=\"fs-id1165135669594\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135176855\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135669594-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y &<& 1 \\\\ y &>& 2x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134544651\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134544652\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y &<& -5 \\\\ y &>& 5x+10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133404494\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133404495\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133404494-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y^2 &<& 25 \\\\ 3x^2-y^2 &>& 12 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135491494\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135491495\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-y^2 &>& -4 \\\\ x^2+y^2 &<& 12 \\end{array}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135474650\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135474651\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135474650-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+3y^2 &>& 16 \\\\ 3x^2-y^2 &<& 1 \\end{array}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135680032\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165135680037\">For the following exercises, graph the inequality.<\/p>\n<div id=\"fs-id1165135680040\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135680041\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y &\\ge& e^x \\\\ y &\\le& \\ln(x)+5 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165133183427\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133183428\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133183427-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} y &\\le& -\\log(x) \\\\ y &\\le& e^x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134185314\">For the following exercises, find the solutions to the nonlinear equations with two variables.<\/p>\n<div id=\"fs-id1165134185317\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134185318\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{4}{x^2}+\\frac{1}{y^2} &=& 24 \\\\ \\frac{5}{x^2}-\\frac{2}{y^2}+4 &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131911076\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131911077\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131911076-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{6}{x^2}-\\frac{1}{y^2} &=& 8 \\\\ \\frac{1}{x^2}-\\frac{6}{y^2} &=& \\frac{1}{8} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135461999\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135462000\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-xy+y^2-2 &=& 0 \\\\ x+3y &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134223178\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134223179\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134223178-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2-xy-2y^2-6 &=& 0 \\\\ x^2+y^2 &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132213505\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132213506\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+4xy-2y^2-6 &=& 0 \\\\ x &=& y+2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134237067\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165134237072\">For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.<\/p>\n<div id=\"fs-id1165135239654\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135239655\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135239654-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} xy &<& 1 \\\\ y &>& \\sqrt{x} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165134181692\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134181693\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x^2+y &<& 3 \\\\ y &>& 2x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134183638\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1165133007623\">For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.<\/p>\n<div id=\"fs-id1165133007628\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133007629\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133007628-solution\">55<\/a><span class=\"os-divider\">. <\/span>Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135602212\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135602213\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>The squares of two numbers add to 360. The second number is half the value of the first number squared. What are the numbers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135602219\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135602220\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135602219-solution\">57<\/a><span class=\"os-divider\">. <\/span>A laptop company has discovered their cost and revenue functions for each day: [latex]C(x)=3x^2-10x+200[\/latex] and [latex]R(x)=-2x^2+100x+50.[\/latex] If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135695987\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135695988\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>A cell phone company has the following cost and revenue functions: [latex]C(x)=8x^2-600x+21,500[\/latex] and [latex]R(x)=-3x^2+480x.[\/latex] What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\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-214","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/214","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":10,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/214\/revisions"}],"predecessor-version":[{"id":1696,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/214\/revisions\/1696"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/214\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=214"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=214"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=214"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}