{"id":213,"date":"2025-04-09T17:28:43","date_gmt":"2025-04-09T17:28:43","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-2-systems-of-linear-equations-three-variables-college-algebra-2e-openstax\/"},"modified":"2025-09-03T19:14:49","modified_gmt":"2025-09-03T19:14:49","slug":"7-2-systems-of-linear-equations-three-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-2-systems-of-linear-equations-three-variables\/","title":{"raw":"7.2 Systems of Linear Equations: Three Variables","rendered":"7.2 Systems of Linear Equations: Three 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_4f2b4755-b59c-4f4f-b4c3-e97938c93fcd\" 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 systems of three equations in three variables.<\/li>\r\n \t<li>Identify inconsistent systems of equations containing three variables.<\/li>\r\n \t<li>Express the solution of a system of dependent equations containing three variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Figure_09_02_005\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n\r\n[caption id=\"attachment_1697\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1697\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-300x225.webp\" alt=\"\" width=\"300\" height=\"225\" \/> Figure 1 (credit: \"Elembis,\" Wikimedia Commons)[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137936545\">Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?<\/p>\r\n<p id=\"fs-id1165135347257\">Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.<\/p>\r\n\r\n<section id=\"fs-id1165134589448\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Systems of Three Equations in Three Variables<\/h2>\r\n<p id=\"fs-id1165134032275\">In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution [latex] (x, y, z), [\/latex] which we call an ordered triple. A system in upper triangular form looks like the following:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} Ax+By+Cz &amp;=&amp; D \\\\ Ey+Fz &amp;=&amp; G \\\\ Hz &amp;=&amp; K \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165135332967\">The third equation can be solved for z and then we back-substitute to find <em>y<\/em> and <em>x<\/em>. To write the system in upper triangular form, we can perform the following operations:<\/p>\r\n\r\n<ol id=\"fs-id1165137735575\" type=\"1\">\r\n \t<li>Interchange the order of any two equations.<\/li>\r\n \t<li>Multiply both sides of an equation by a nonzero constant.<\/li>\r\n \t<li>Add a nonzero multiple of one equation to another equation.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135686641\">The <strong>solution set<\/strong> to a three-by-three system is an ordered triple [latex] {(x, y, z)}. [\/latex] Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.<\/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=\"\">Number of Possible Solutions <\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFigure 2 and Figure 3 illustrate possible solution scenarios for three-by-three systems.\r\n<ul>\r\n \t<li>Systems that have a single solution are those which, after elimination, result in a <strong>solution set<\/strong> consisting of an ordered triple [latex] {(x, y, z)}. [\/latex] Graphically, the ordered triple defines a point that is the intersection of three planes in space.<\/li>\r\n \t<li>Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as [latex] 0=0. [\/latex] Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space.<\/li>\r\n \t<li>Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex] 3=0. [\/latex] Graphically, a system with no solution is represented by three planes with no point in common.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1045\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1045\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-300x147.webp\" alt=\"\" width=\"300\" height=\"147\" \/> Figure 2. (a)Three planes intersect at a single point, representing a three-by-three system with a single solution. (b) Three planes intersect in a line, representing a three-by-three system with infinite solutions.[\/caption]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1046\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1046 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-300x116.webp\" alt=\"\" width=\"300\" height=\"116\" \/> Figure 3. All three figures represent three-by-three systems with no solution. (a) The three planes intersect with each other, but not at a common point. (b) Two of the planes are parallel and intersect with the third plane, but not with each other. (c) All three planes are parallel, so there is no point of intersection.[\/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 1: Determining Whether an Ordered Triple Is a Solution to a System<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine whether the ordered triple [latex] (3, -2, 1) [\/latex] is a solution to the system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 2 \\\\ 6x-4y+5z &amp;=&amp; 31 \\\\ 5x+2y+2z &amp;=&amp; 13 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We will check each equation by substituting in the values of the ordered triple for <em>x<\/em>, <em>y<\/em>, and <em>z<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rrrr} x+y+z &amp;=&amp; 2 \\\\ (3)+(-2)+(1) &amp;=&amp; 2 \\\\ &amp;&amp; \\text{True} \\\\ \\\\ 6x-4y+5z &amp;=&amp; 31 \\\\ 6(3)-4(-2)+5(1) &amp;=&amp; 31 \\\\ &amp;&amp; \\text{True} \\\\ \\\\ 5x+2y+2z &amp;=&amp; 13 \\\\ 5(3)+2(-2)+2(1) &amp;=&amp; 13 \\\\ &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\nThe ordered triple [latex] (3, -2, 1) [\/latex] is indeed a solution to the system.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/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 linear system of three equations, solve for three unknowns.<\/strong>\r\n<ol>\r\n \t<li>Pick any pair of equations and solve for one variable.<\/li>\r\n \t<li>Pick another pair of equations and solve for the same variable.<\/li>\r\n \t<li>You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.<\/li>\r\n \t<li>Back-substitute known variables into any one of the original equations and solve for the missing variable.<\/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: Solving a System of Three Equations in Three Variables by Elimination<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a solution to the following system:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2y+3x &amp;=&amp; 9 &amp;&amp; (1) \\\\ -x+3y-z &amp;=&amp; -6 &amp;&amp; (2) \\\\ 2x-5y+5z &amp;=&amp; 17 &amp;&amp; (3) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>There will always be several choices as to where to begin, but the most obvious first step here is to eliminate <em>x<\/em> by adding equations (1) and (2).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2y+3x &amp;=&amp; 9 &amp;&amp; (1) \\\\ -x+3y-z &amp;=&amp; -6 &amp;&amp; 92) \\\\ \\hline y+2z &amp;=&amp; 3 &amp;&amp; (4) \\end{array} [\/latex]<\/p>\r\nThe second step is multiplying equation (1) by [latex] -2 [\/latex] and adding the result to equation (3). These two steps will eliminate the variable <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -2x+4y-6z &amp;=&amp; -18 &amp;&amp; (1) \\ \\text{multiplied by } -2 \\\\ 2x-5y+5z &amp;=&amp; 17 &amp;&amp; (3) \\\\ \\hline -y-z &amp;=&amp; -1 &amp;&amp; (5) \\end{array} [\/latex]<\/p>\r\nIn equations (4) and (5), we have created a new two-by-two system. We can solve for z by adding the two equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y+2z &amp;=&amp; 3 &amp;&amp; (4) \\\\ -y-z &amp;=&amp; -1 &amp;&amp; (5) \\\\ \\hline z &amp;=&amp; 2 &amp;&amp; (6) \\end{array} [\/latex]<\/p>\r\nChoosing one equation from each new system, we obtain the upper triangular form:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2y+3z &amp;=&amp; 9 &amp;&amp; (1) \\\\ y+2z &amp;=&amp; 3 &amp;&amp; (4) \\\\ z &amp;+&amp; 2 &amp;&amp; (6) \\end{array} [\/latex]<\/p>\r\nNext, we back-substitute [latex] z=2 [\/latex] into equation (4) and solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y+2(2) &amp;=&amp; 3 \\\\ y+4 &amp;=&amp; 3 \\\\ y &amp;=&amp; -1 \\end{array} [\/latex]<\/p>\r\nFinally, we can back-substitute [latex] z=2 [\/latex] and [latex] y=-1 [\/latex] into equation (1). This will yield the solution for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2(-1)+3(2) &amp;=&amp; 9 \\\\ x+2+6 &amp;=&amp; 9 \\\\ x &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\nThe solution is the ordered triple [latex] (1, -1, 2). [\/latex] See Figure 4.\r\n\r\n[caption id=\"attachment_1047\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1047\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-300x250.webp\" alt=\"\" width=\"300\" height=\"250\" \/> Figure 4[\/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\">Example 3: Solving a Real-World Problem Using a System of Three Equations in Three Variables<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn the problem posed at the beginning of the section, Jordi invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. Jordi invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670. How much did he invest in each type of fund?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x &amp;=&amp; \\text{amount invested in money-market fund} \\\\ y &amp;=&amp; \\text{amoung invested in municipal bonds} \\\\ z &amp;=&amp; \\text{amount invsted in mutual funds} \\end{array} [\/latex]<\/p>\r\nThe first equation indicates that the sum of the three principal amounts is $12,000.\r\n<p style=\"text-align: center;\">[latex] x+y+z=12,000 [\/latex]<\/p>\r\nWe form the second equation according to the information that Jordi invested $4,000 more in mutual funds than he invested in municipal bonds.\r\n<p style=\"text-align: center;\">[latex] z=y+4,000 [\/latex]<\/p>\r\nThe third equation shows that the total amount of interest earned from each fund equals $670.\r\n<p style=\"text-align: center;\">[latex] 0.03x+0.04y+0.07z=670 [\/latex]<\/p>\r\nThen, we write the three equations as a system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 12,000 \\\\ -y+z &amp;=&amp; 4,000 \\\\ 0.03x+0.04y+0.07z &amp;=&amp; 670 \\end{array} [\/latex]<\/p>\r\nTo make the calculations simpler, we can multiply the third equation by 100. Thus,\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 12,000 &amp;&amp; (1) \\\\ -y+z &amp;=&amp; 4,000 &amp;&amp; (2) \\\\ 3x+4y+7z &amp;=&amp; 67,000 &amp;&amp; (3) \\end{array} [\/latex]<\/p>\r\nStep 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up.\r\n<p style=\"text-align: center;\">[latex \\begin{array}{rll} x+y+z &amp;=&amp; 12,000 \\\\ 3x+4y+7z &amp;=&amp; 67,000 \\\\ -y+z &amp;=&amp; 4,000 \\end{array}] [\/latex]<\/p>\r\nStep 2. Multiply equation (1) by [latex] -3 [\/latex] and add to equation (2). Write the result as row 2.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 12,000 \\\\ y+4z &amp;=&amp; 31,000 \\\\ -y+z &amp;=&amp; 4,000 \\end{array} [\/latex]<\/p>\r\nStep 3. Add equation (2) to equation (3) and write the result as equation (3).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 12,000 \\\\ y+4z &amp;=&amp; 31,000 \\\\ 5z &amp;=&amp; 35,000 \\end{array} [\/latex]<\/p>\r\nStep 4. Solve for <em>z<\/em> in equation (3). Back-substitute that value in equation (2) and solve for <em>y<\/em>. Then, back-substitute the values for <em>z<\/em> and <em>y<\/em> into equation (1) and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 5z &amp;=&amp; 35,000 \\\\ z &amp;=&amp; 7,000 \\\\ \\\\ \\\\ y+4(7,000) &amp;=&amp; 31,000 \\\\ y &amp;=&amp; 3,000 \\\\ \\\\ \\\\ x+3,000+7,000 &amp;=&amp; 12,000 \\\\ x &amp;=&amp; 2,000 \\end{array} [\/latex]<\/p>\r\nJordi invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds.\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 #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of equations in three variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y-2z &amp;=&amp; -1\\\\ 3x-3y-z &amp;=&amp; 5 \\\\ x-2y+3z &amp;=&amp; 6 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134162165\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Three Variables<\/h2>\r\n<p id=\"fs-id1165135541808\">Just as with systems of equations in two variables, we may come across an <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">inconsistent system<\/span> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as [latex] 3=7 [\/latex] or some other contradiction.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Solving an Inconsistent System of Three Equations in Three Variables<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} cx-3y+z &amp;=&amp; 4 &amp;&amp; (1) \\\\ -x+2y-5z &amp;=&amp; 3 &amp;&amp; (2) \\\\ 5x-13y+13z &amp;=&amp; 8 &amp;&amp; (3) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Looking at the coefficients of <em>x<\/em> we can see that we can eliminate <em>x<\/em> by adding equation (1) to equation (2).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-3y+z &amp;=&amp; 4 &amp;&amp; (1) \\\\ -x+2y-5z &amp;=&amp; 3 &amp;&amp; (2) \\\\ \\hline -y-4z &amp;=&amp; 7 &amp;&amp; (4) \\end{array} [\/latex]<\/p>\r\nNext, we multiply equation (1) by [latex] -5 [\/latex] and add it to equation (3).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -5x+15y-5z &amp;=&amp; -20 &amp;&amp; (1) \\ \\text{multiplied by } -5 \\\\ 5x-13y+13z &amp;-&amp; 8 &amp;&amp; (3) \\\\ \\hline 2y+8z &amp;=&amp; -12 &amp;&amp; (5) \\end{array} [\/latex]<\/p>\r\nThen, we multiply equation (4) by 2 and add it to equation (5).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -2y-8z &amp;=&amp; 14 &amp;&amp; (4) \\ \\text{multiplied by } 2 \\\\ 2y+8z &amp;=&amp; -12 &amp;&amp; (5) \\\\ \\hline 0 &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\nThe final equation [latex] 0=2 [\/latex] is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.\r\n<h3>Analysis<\/h3>\r\nIn this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165135315557\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137678319\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of three equations in three variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 2 \\\\ y-3z &amp;=&amp; 1 \\\\ 2x+y+5z &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165134279181\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Expressing the Solution of a System of Dependent Equations Containing Three Variables<\/h2>\r\n<p id=\"fs-id1165134279186\">We know from working with systems of equations in two variables that a <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">dependent system<\/span> of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding the Solution to a Dependent System of Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the solution to the given system of three equations in three variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y-3z &amp;=&amp; 0 &amp;&amp; (1) \\\\ 4x+2y-6z &amp;=&amp; 0 &amp;&amp; (2) \\\\ x-y+z &amp;=&amp; 0 &amp;&amp; (3) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, we can multiply equation (1) by [latex] -2 [\/latex] and add it to equation (2).\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -4x-2y+6z &amp;=&amp; 0 &amp;&amp; \\text{equation (1) multiplied by } -2 \\\\ 4x+2y-6z &amp;=&amp; 0 &amp;&amp; (2) \\\\ \\hline 0=0 \\end{array} [\/latex]<\/p>\r\nWe do not need to proceed any further. The result we get is an identity, [latex] 0=0 [\/latex] which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by [latex] -2 [\/latex] and adding it to equation (1). We then perform the same steps as above and find the same result, [latex] 0=0. [\/latex]\r\n\r\nWhen a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y-3z &amp;=&amp; 0 \\\\ x-y+z &amp;=&amp; 0 \\\\ \\hline 3x-2z &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\nWe then solve the resulting equation for <em>z<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3x-2z &amp;=&amp; 0 \\\\ z &amp;=&amp; \\frac{3}{2}x \\end{array} [\/latex]<\/p>\r\nWe back-substitute the expression for <em>z<\/em> into one of the equations and solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y-3\\left(\\frac{3}{2}x\\right) &amp;=&amp; 0 \\\\ 2x+y-\\frac{9}{2}x &amp;=&amp; 0 \\\\ y &amp;=&amp; \\frac{9}{2}x-2x \\\\ y &amp;=&amp; \\frac{5}{2}x \\end{array} [\/latex]<\/p>\r\nSo the general solution is [latex] \\left(x, \\frac{5}{2}x, \\frac{3}{2}x\\right). [\/latex] In this solution, <em>x<\/em> can be any real number. The values of <em>y<\/em> and <em>z<\/em> are dependent on the value selected for <em>x<\/em>.\r\n<h3>Analysis<\/h3>\r\nAs shown in Figure 5, two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.\r\n\r\n[caption id=\"attachment_1048\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1048\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-300x238.webp\" alt=\"\" width=\"300\" height=\"238\" \/> Figure 5[\/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>Q: Does the generic solution to a dependent system always have to be written in terms of x? <\/strong>\r\n\r\n<em data-effect=\"italics\">A: No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed x and y. <\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\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\r\nSolve the following system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+y+z &amp;=&amp; 7 \\\\ 3x-2y-z &amp;=&amp; 4 \\\\ x+6y+5z &amp;=&amp; 24 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/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 systems of equations in three variables.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=3RbVSvvRyeI\">Ex 1: System of Three Equations with Three Unknowns Using Elimination<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=r6htz3gaHZ0\">Ex 2: System of Three Equations with Three Unknowns Using Elimination<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_09_02_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135191971\" data-type=\"problem\"><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.2 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137428022\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137581426\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137581432\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135582116\" 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-id1165137581432-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can a linear system of three equations have exactly two solutions? Explain why or why not\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134116743\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134116745\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135421454\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135421456\" 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-id1165135421454-solution\">3<\/a><span class=\"os-divider\">. <\/span>If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135354941\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135354943\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Using the method of addition, is there only one way to solve the system?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137611460\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133238406\" 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-id1165137611460-solution\">5<\/a><span class=\"os-divider\">. <\/span>Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134205883\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165135161479\">For the following exercises, determine whether the ordered triple given is the solution to the system of equations.<\/p>\r\n\r\n<div id=\"fs-id1165135161483\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135432898\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x-6y+6z &amp;=&amp; -12 \\\\ x+4y+5z &amp;=&amp; -1 \\\\ -x+2y+3z &amp;=&amp; -1 \\end{array} [\/latex] and [latex] (0, 1, -1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165132079296\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132079298\" 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-id1165132079296-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 6x-y+3z&amp;=&amp; 6 \\\\ 3x+5y+2z &amp;\\&amp; 0 \\\\ x+y &amp;=&amp; 0 \\end{array} [\/latex] and [latex] (3, -3, -5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134430325\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134430327\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 6x-7y+z &amp;=&amp; 2 \\\\ -x-y+3z &amp;+&amp; 4 \\\\ 2x+y-z &amp;=&amp; 1 \\end{array} [\/latex] and [latex] (4, 2,-6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133306920\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133306922\" 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-id1165133306920-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x-y &amp;=&amp; 0 \\\\ x-z &amp;=&amp; 5 \\\\ x-y+z &amp;=&amp; -1 \\end{array} [\/latex] and [latex] (4, 4, -1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135577081\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135553486\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x-y+2z &amp;=&amp; 3 \\\\ 5x+8y-3z &amp;=&amp; 4 \\\\ -x+3y-5z &amp;=&amp; -5 \\end{array} [\/latex] and [latex] (4, 1, -7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137823121\">For the following exercises, solve each system by elimination.<\/p>\r\n\r\n<div id=\"fs-id1165137823124\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137823126\" 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-id1165137823124-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-4y+2z &amp;=&amp; -15 \\\\ 2x+4y+z &amp;=&amp; 16 \\\\ 2x+3y+5z &amp;=&amp; 20 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137784201\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137784203\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-2y+3z &amp;=&amp; 20 \\\\ 2x-4y-3z &amp;=&amp; -9 \\\\ x+6y-8z &amp;=&amp; 21 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137898682\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898684\" 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-id1165137898682-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x+2y+4z &amp;=&amp; 9 \\\\ -3x+2y+z &amp;=&amp; 10 \\\\ 4x-3y+5z &amp;=&amp; -3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135609190\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135609192\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 4x-3y+5z &amp;=&amp; 31 \\\\ -x+2y+4z &amp;=&amp; 20 \\\\ x+5y-2z &amp;=&amp; -29 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134089381\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134089383\" 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-id1165134089381-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-2y+3z &amp;=&amp; 4 \\\\ -4x+6y-7z &amp;=&amp; -1 \\\\ 3x+2y-z &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135702605\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135702607\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 4x+6y+9z &amp;=&amp; 0 \\\\ -5x+2y-6z &amp;=&amp; 3 \\\\ 7x-4y+3z &amp;=&amp; -3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134280366\">For the following exercises, solve each system by Gaussian elimination.<\/p>\r\n\r\n<div id=\"fs-id1165133360321\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133360323\" 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-id1165133360321-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x-y+3z &amp;=&amp; 17 \\\\ -5x+4y-2z &amp;=&amp; -46 \\\\ 2y+5z&amp;=&amp; -7 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133046785\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133046787\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-6y+3z &amp;=&amp; 50 \\\\ -x+4y &amp;=&amp; 10 \\\\ 2x-z &amp;=&amp; 10 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131907322\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131907324\" 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-id1165131907322-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x+3y-6z &amp;=&amp; 1 \\\\ -4x=6y+12z &amp;=&amp; -2 \\\\ x+2y+5z &amp;=&amp; 10 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135389856\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135389858\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 4x+6y-2z &amp;=&amp; 8 \\\\ 6x+9y-3z &amp;=&amp; 12 \\\\ -2x-3y+z &amp;=&amp; -4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135633857\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134216211\" 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-id1165135633857-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x+3y-4z &amp;=&amp; 5 \\\\ -3x+2y+z &amp;=&amp; 11 \\\\ -x+5y+3z &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135707939\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135707941\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 10x+2y-14z &amp;=&amp; 8 \\\\ -x-2y-4z &amp;=&amp; -1 \\\\ -12x-6y+6z &amp;=&amp; -12 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134368156\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134368158\" 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-id1165134368156-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y+z &amp;=&amp; 14 \\\\ 2y+3z &amp;=&amp; -14 \\\\ -16y-24z &amp;=&amp; -112 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134072235\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135620859\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-3y+4z &amp;=&amp; -1 \\\\ -4x+2y-3z &amp;=&amp; 0 \\\\ -x+5y+7z &amp;=&amp; -11 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137938397\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137938399\" 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-id1165137938397-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y+z &amp;=&amp; 0 \\\\ 2x-y+3z &amp;=&amp; 0 \\\\ x-z &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135602254\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135602256\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x+2y-5z &amp;=&amp; 6 \\\\ 5x-4y+3z &amp;=&amp; -12 \\\\ 4x+5y-2z &amp;=&amp; 15 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134037555\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134037557\" 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-id1165134037555-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y+z &amp;=&amp; 0 \\\\ 2x-y+3z &amp;=&amp; 0 \\\\ x-z &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133391507\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133391509\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-\\frac{1}{2}y-z &amp;=&amp; -\\frac{1}{2} \\\\ 4x+z &amp;=&amp; 3 \\\\ -x+\\frac{3}{2}y &amp;=&amp; \\frac{5}{2} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135316107\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135316109\" 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-id1165135316107-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 6x-5y+6z &amp;=&amp; 38 \\\\ \\frac{1}{5}x-\\frac{1}{2}y+\\frac{3}{5}z &amp;=&amp; 1 \\\\ -4x-\\frac{3}{2}y-z &amp;=&amp; -74 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135516682\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135516684\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{2}x-\\frac{1}{5}y+\\frac{2}{5}z &amp;=&amp; -\\frac{13}{10} \\\\ \\frac{1}{4}x-\\frac{2}{5}y-\\frac{1}{5}z &amp;=&amp; -\\frac{7}{20} \\\\ -\\frac{1}{2}x-\\frac{3}{4}y-\\frac{1}{2}z &amp;=&amp; -\\frac{5}{4} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133358779\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135253693\" 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-id1165133358779-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -\\frac{1}{3}x-\\frac{1}{2}y-\\frac{1}{4}z &amp;=&amp; \\frac{3}{4} \\\\ -\\frac{1}{2}x-\\frac{1}{4}y-\\frac{1}{2}z &amp;=&amp; 2 \\\\ -\\frac{1}{4}x-\\frac{3}{4}y-\\frac{1}{2}z &amp;=&amp; -\\frac{1}{2} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132005229\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132005232\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{2}x-\\frac{1}{4}y+\\frac{3}{4}z &amp;=&amp; 0 \\\\ \\frac{1}{4}x+\\frac{1}{10}y+\\frac{2}{5}z &amp;=&amp; -2 \\\\ \\frac{1}{8}x+\\frac{1}{5}y-\\frac{1}{8}z &amp;=&amp; 2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199541\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199543\" 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-id1165134199541-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{4}{5}x-\\frac{7}{8}y+\\frac{1}{2}z &amp;=&amp; 1 \\\\ -\\frac{4}{5}x-\\frac{3}{4}y+\\frac{1}{3}z &amp;=&amp; -8 \\\\ -\\frac{2}{5}x-\\frac{7}{8}y+\\frac{1}{2}z &amp;=&amp; -5 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137896940\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137896942\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -\\frac{1}{3}x-\\frac{1}{8}y+\\frac{1}{6}z &amp;=&amp; -\\frac{4}{3} \\\\ -\\frac{2}{3}x-\\frac{7}{8}y+\\frac{1}{3}z &amp;=&amp; -\\frac{23}{3} \\\\ -\\frac{1}{3}x-\\frac{5}{8}y+\\frac{5}{6}z &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135485720\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485722\" 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-id1165135485720-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -\\frac{1}{4}x-\\frac{5}{4}y+\\frac{5}{2}z &amp;=&amp; -5 \\\\ -\\frac{1}{2}x-\\frac{5}{2}y+\\frac{5}{4}z &amp;=&amp; \\frac{55}{12} \\\\ -\\frac{1}{3}x-\\frac{1}{3}y+\\frac{1}{3}z &amp;=&amp; \\frac{5}{3} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134254135\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134254137\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{40}x+\\frac{1}{60}y+\\frac{1}{80}z &amp;=&amp; \\frac{1}{100} \\\\ -\\frac{1}{2}x-\\frac{1}{3}y-\\frac{1}{4}z &amp;=&amp; -\\frac{1}{5} \\\\ \\frac{3}{8}x+\\frac{3}{12}y+\\frac{3}{16}z &amp;=&amp; \\frac{3}{20} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135417018\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135417020\" 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-id1165135417018-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.1x-0.2y+0.3z &amp;=&amp; 2 \\\\ 0.5x=0.1y+0.4z &amp;=&amp; 8 \\\\ 0.7x-0.2y+0.3z &amp;=&amp; 8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134277203\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134277205\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.2x+0.1y-0.3z &amp;=&amp; 0.2 \\\\ 0.8x+0.4y-1.2z &amp;=&amp; 0.1 \\\\ 1.6x+0.8y-2.4z &amp;=&amp; 0.2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135450286\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135450288\" 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-id1165135450286-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 1.1x+0.7y-3.1z &amp;=&amp; -1.79 \\\\ 2.1x+0.5y-1.6z &amp;=&amp; -0.13 \\\\ 0.5x+0.4y-0.5z &amp;=&amp; -0.07 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135202434\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133134690\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.5x-0.5+0.5z &amp;=&amp; 10 \\\\ 0.2x-0.2y+0.2z &amp;=&amp; 4 \\\\ 0.1x-0.1y+0.1z &amp;=&amp; 2 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133194612\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133194614\" 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-id1165133194612-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.1x+0.2y+0.3z &amp;=&amp; 0.37 \\\\ 0.1x-0.2y-0.3z &amp;=&amp; -0.27 \\\\ 0.5x-0.1y-0.3z &amp;=&amp; -0.03 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135672780\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135672782\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.5x-0.5y-0.3z &amp;=&amp; 0.13 \\\\ 0.4x-0.1y-0.3z &amp;=&amp; 0.11 \\\\ 0.2x-0.8y-0.9z &amp;=&amp; -0.32 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135640563\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135640565\" 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-id1165135640563-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.5x+0.2y-0.3z &amp;=&amp; 1 \\\\ 0.4x-0.6y+0.7z &amp;=&amp; 0.8 \\\\ 0.3x-0.1y-0.9z &amp;=&amp; 0.6 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133393400\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133393403\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.3x+0.3y+0.5x &amp;=&amp; 0.6 \\\\ 0.4x+0.4y+0.4z &amp;=&amp; 1.8 \\\\ 0.4x+0.2y+0.1z &amp;=&amp; 1.6 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132005156\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132005158\" 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-id1165132005156-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.8x+0.8y+0.8z &amp;=&amp; 2.4 \\\\ 0.3x-0.5y+0.2z &amp;=&amp; 0 \\\\ 0.1x+0.2y+0.3z &amp;=&amp; 0.6 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165133294287\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165135537327\">For the following exercises, solve the system for x, y, and z.<\/p>\r\n\r\n<div id=\"fs-id1165133441059\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133441061\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y+z &amp;=&amp; 3 \\\\ \\frac{x-1}{2}+\\frac{y-3}{2}+\\frac{z+1}{2} &amp;=&amp; 0 \\\\ \\frac{x-2}{3}+\\frac{y+4}{3}+\\frac{z-3}{3} &amp;=&amp; \\frac{2}{3} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135539006\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135539008\" 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-id1165135539006-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-3y-\\frac{z+1}{2} &amp;=&amp; \\frac{1}{2} \\\\ 6x+\\frac{y-9}{2}+2z &amp;=&amp; -3 \\\\ \\frac{x+8}{2}-4y+z &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133320194\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133320196\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{x+4}{7}-\\frac{y-1}{6}+\\frac{z+2}{3} &amp;=&amp; 1 \\\\ \\frac{x-2}{4}+\\frac{y+1}{8}-\\frac{z+8}{12} &amp;=&amp; 0 \\\\ \\frac{x+6}{3}-\\frac{y+2}{3}+\\frac{z+4}{2} &amp;=&amp; 3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132035881\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132035883\" 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-id1165132035881-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{x-3}{6}+\\frac{y+2}{2}-\\frac{z-3}{3} &amp;=&amp; 2 \\\\ \\frac{x+2}{4}+\\frac{y-5}{2}+\\frac{z+4}{2} &amp;=&amp; 1 \\\\ \\frac{x+6}{2}-\\frac{y-3}{2}+z+1 &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135344036\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135344039\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{x-1}{3}+\\frac{y+3}{4}+\\frac{z+2}{6} &amp;=&amp; 1 \\\\ 4x+3y-2z &amp;=&amp; 11 \\\\ 0.02x+0.015y-0.01z &amp;=&amp; 0.065 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135258841\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165134053995\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134053997\" 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-id1165134053995-solution\">51<\/a><span class=\"os-divider\">. <\/span>Three even numbers sum up to 108. The smaller is half the larger and the middle number is [latex] \\frac{3}{4} [\/latex] the larger. What are the three numbers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134190578\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134190580\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132972738\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132972740\" 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-id1165132972738-solution\">53<\/a><span class=\"os-divider\">. <\/span>At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132972757\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132972759\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>The Aurora animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133309239\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133309241\" 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-id1165133309239-solution\">55<\/a><span class=\"os-divider\">. <\/span>Your roommate, Shani, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate\u2019s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134072281\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134072283\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Your roommate, John, offered to buy household supplies for you and your other roommate. You live near the border of three states, each of which has a different sales tax. The total amount of money spent was $100.75. Your supplies were bought with 5% tax, John\u2019s with 8% tax, and your third roommate\u2019s with 9% sales tax. The total amount of money spent without taxes is $93.50. If your supplies before tax were $1 more than half of what your third roommate\u2019s supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134072299\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137394065\" 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-id1165134072299-solution\">57<\/a><span class=\"os-divider\">. <\/span>Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137394083\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137394085\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>At a carnival at the Aurora Mall, $2,914.25 in receipts were taken at the end of the day. The cost of a child\u2019s ticket was $20.50, an adult ticket was $29.75, and a senior citizen ticket was $15.25. There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135705887\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135705889\" 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-id1165135705887-solution\">59<\/a><span class=\"os-divider\">. <\/span>Skool Daze, an Aurora, CO, band, sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134231621\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134231623\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>In a bag, a child has 325 coins worth $19.50. There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134231632\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134231635\" 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-id1165134231632-solution\">61<\/a><span class=\"os-divider\">. <\/span>Last year, at Stevinson Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $140,000. This year, due to inflation, the same cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year\u2019s Jeep was $7,000 less than the price of last year\u2019s BMW, what was the price of each of the three cars last year?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131950632\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131950634\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>When his youngest child moved out, Deandre sold his home and made three investments using gains from the sale. He invested $80,500 into three accounts, one that paid 4% simple interest, one that paid [latex] 3 \\frac{1}{8}\\% [\/latex] simple interest, and one that paid [latex] 2 \\frac{1}{2}\\% [\/latex] simple interest. He earned $2,670 interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134138635\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134138637\" 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-id1165134138635-solution\">63<\/a><span class=\"os-divider\">. <\/span>You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134554932\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134554934\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>An entrepreneur sells a portion of their business for one hundred thousand dollars and invests it all in three accounts for one year. The first account pays 4% compounded annually, the second account pays 3% compounded annually, and the third account pays 2% compounded annually. After one year, the entrepreneur earns $3,650 in interest. If they invested five times the money in the account that pays 4% compared to 3%, how much did they invest in each account?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134381545\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134381547\" 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-id1165134381545-solution\">65<\/a><span class=\"os-divider\">. <\/span>The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world\u2019s consumed oil. The United States consumed 0.7% more than four times China\u2019s consumption. The United States consumed 5% more than triple Japan\u2019s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137404768\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137404770\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced 31.4% of the world\u2019s produced oil. Saudi Arabia and the United States combined for 22.1% of the world\u2019s production, and Saudi Arabia produced 2% more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134226686\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134226688\" 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-id1165134226686-solution\">67<\/a><span class=\"os-divider\">. <\/span>The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported 1.8% more from Saudi Arabia than they did from Mexico, and 1.7% more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135367795\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135367797\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for 64% of the United States oil production. The Gulf of Mexico and Texas combined for 47% of oil production. Texas produced 3% more than Alaska. What percent of United States oil production came from these regions?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135367810\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135367812\" 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-id1165135367810-solution\">69<\/a><span class=\"os-divider\">. <\/span>At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134116842\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134116844\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Meat consumption in the United States can be broken into three categories: red meat, poultry, and fish. If fish makes up 4% less than one-quarter of poultry consumption, and red meat consumption is 18.2% higher than poultry consumption, what are the percentages of meat consumption?\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_4f2b4755-b59c-4f4f-b4c3-e97938c93fcd\" 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 systems of three equations in three variables.<\/li>\n<li>Identify inconsistent systems of equations containing three variables.<\/li>\n<li>Express the solution of a system of dependent equations containing three variables.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Figure_09_02_005\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_1697\" aria-describedby=\"caption-attachment-1697\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1697\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-300x225.webp\" alt=\"\" width=\"300\" height=\"225\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-300x225.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-65x49.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-225x169.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1-350x263.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-1.webp 599w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1697\" class=\"wp-caption-text\">Figure 1 (credit: &#8220;Elembis,&#8221; Wikimedia Commons)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936545\">Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?<\/p>\n<p id=\"fs-id1165135347257\">Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.<\/p>\n<section id=\"fs-id1165134589448\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Systems of Three Equations in Three Variables<\/h2>\n<p id=\"fs-id1165134032275\">In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution [latex](x, y, z),[\/latex] which we call an ordered triple. A system in upper triangular form looks like the following:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} Ax+By+Cz &=& D \\\\ Ey+Fz &=& G \\\\ Hz &=& K \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135332967\">The third equation can be solved for z and then we back-substitute to find <em>y<\/em> and <em>x<\/em>. To write the system in upper triangular form, we can perform the following operations:<\/p>\n<ol id=\"fs-id1165137735575\" type=\"1\">\n<li>Interchange the order of any two equations.<\/li>\n<li>Multiply both sides of an equation by a nonzero constant.<\/li>\n<li>Add a nonzero multiple of one equation to another equation.<\/li>\n<\/ol>\n<p id=\"fs-id1165135686641\">The <strong>solution set<\/strong> to a three-by-three system is an ordered triple [latex]{(x, y, z)}.[\/latex] Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.<\/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=\"\">Number of Possible Solutions <\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Figure 2 and Figure 3 illustrate possible solution scenarios for three-by-three systems.<\/p>\n<ul>\n<li>Systems that have a single solution are those which, after elimination, result in a <strong>solution set<\/strong> consisting of an ordered triple [latex]{(x, y, z)}.[\/latex] Graphically, the ordered triple defines a point that is the intersection of three planes in space.<\/li>\n<li>Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as [latex]0=0.[\/latex] Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space.<\/li>\n<li>Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex]3=0.[\/latex] Graphically, a system with no solution is represented by three planes with no point in common.<\/li>\n<\/ul>\n<figure id=\"attachment_1045\" aria-describedby=\"caption-attachment-1045\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1045\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-300x147.webp\" alt=\"\" width=\"300\" height=\"147\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-300x147.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-65x32.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-225x110.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2-350x171.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-2.webp 635w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1045\" class=\"wp-caption-text\">Figure 2. (a)Three planes intersect at a single point, representing a three-by-three system with a single solution. (b) Three planes intersect in a line, representing a three-by-three system with infinite solutions.<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1046\" aria-describedby=\"caption-attachment-1046\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1046 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-300x116.webp\" alt=\"\" width=\"300\" height=\"116\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-300x116.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-65x25.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-225x87.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3-350x135.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-3.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1046\" class=\"wp-caption-text\">Figure 3. All three figures represent three-by-three systems with no solution. (a) The three planes intersect with each other, but not at a common point. (b) Two of the planes are parallel and intersect with the third plane, but not with each other. (c) All three planes are parallel, so there is no point of intersection.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Determining Whether an Ordered Triple Is a Solution to a System<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine whether the ordered triple [latex](3, -2, 1)[\/latex] is a solution to the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 2 \\\\ 6x-4y+5z &=& 31 \\\\ 5x+2y+2z &=& 13 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We will check each equation by substituting in the values of the ordered triple for <em>x<\/em>, <em>y<\/em>, and <em>z<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rrrr} x+y+z &=& 2 \\\\ (3)+(-2)+(1) &=& 2 \\\\ && \\text{True} \\\\ \\\\ 6x-4y+5z &=& 31 \\\\ 6(3)-4(-2)+5(1) &=& 31 \\\\ && \\text{True} \\\\ \\\\ 5x+2y+2z &=& 13 \\\\ 5(3)+2(-2)+2(1) &=& 13 \\\\ && \\text{True} \\end{array}[\/latex]<\/p>\n<p>The ordered triple [latex](3, -2, 1)[\/latex] is indeed a solution to the system.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\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 linear system of three equations, solve for three unknowns.<\/strong><\/p>\n<ol>\n<li>Pick any pair of equations and solve for one variable.<\/li>\n<li>Pick another pair of equations and solve for the same variable.<\/li>\n<li>You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.<\/li>\n<li>Back-substitute known variables into any one of the original equations and solve for the missing variable.<\/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: Solving a System of Three Equations in Three Variables by Elimination<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a solution to the following system:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2y+3x &=& 9 && (1) \\\\ -x+3y-z &=& -6 && (2) \\\\ 2x-5y+5z &=& 17 && (3) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>There will always be several choices as to where to begin, but the most obvious first step here is to eliminate <em>x<\/em> by adding equations (1) and (2).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2y+3x &=& 9 && (1) \\\\ -x+3y-z &=& -6 && 92) \\\\ \\hline y+2z &=& 3 && (4) \\end{array}[\/latex]<\/p>\n<p>The second step is multiplying equation (1) by [latex]-2[\/latex] and adding the result to equation (3). These two steps will eliminate the variable <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -2x+4y-6z &=& -18 && (1) \\ \\text{multiplied by } -2 \\\\ 2x-5y+5z &=& 17 && (3) \\\\ \\hline -y-z &=& -1 && (5) \\end{array}[\/latex]<\/p>\n<p>In equations (4) and (5), we have created a new two-by-two system. We can solve for z by adding the two equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y+2z &=& 3 && (4) \\\\ -y-z &=& -1 && (5) \\\\ \\hline z &=& 2 && (6) \\end{array}[\/latex]<\/p>\n<p>Choosing one equation from each new system, we obtain the upper triangular form:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2y+3z &=& 9 && (1) \\\\ y+2z &=& 3 && (4) \\\\ z &+& 2 && (6) \\end{array}[\/latex]<\/p>\n<p>Next, we back-substitute [latex]z=2[\/latex] into equation (4) and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y+2(2) &=& 3 \\\\ y+4 &=& 3 \\\\ y &=& -1 \\end{array}[\/latex]<\/p>\n<p>Finally, we can back-substitute [latex]z=2[\/latex] and [latex]y=-1[\/latex] into equation (1). This will yield the solution for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2(-1)+3(2) &=& 9 \\\\ x+2+6 &=& 9 \\\\ x &=& 1 \\end{array}[\/latex]<\/p>\n<p>The solution is the ordered triple [latex](1, -1, 2).[\/latex] See Figure 4.<\/p>\n<figure id=\"attachment_1047\" aria-describedby=\"caption-attachment-1047\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1047\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-300x250.webp\" alt=\"\" width=\"300\" height=\"250\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-300x250.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-65x54.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-225x188.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4-350x292.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-4.webp 381w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1047\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Solving a Real-World Problem Using a System of Three Equations in Three Variables<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In the problem posed at the beginning of the section, Jordi invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. Jordi invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670. How much did he invest in each type of fund?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x &=& \\text{amount invested in money-market fund} \\\\ y &=& \\text{amoung invested in municipal bonds} \\\\ z &=& \\text{amount invsted in mutual funds} \\end{array}[\/latex]<\/p>\n<p>The first equation indicates that the sum of the three principal amounts is $12,000.<\/p>\n<p style=\"text-align: center;\">[latex]x+y+z=12,000[\/latex]<\/p>\n<p>We form the second equation according to the information that Jordi invested $4,000 more in mutual funds than he invested in municipal bonds.<\/p>\n<p style=\"text-align: center;\">[latex]z=y+4,000[\/latex]<\/p>\n<p>The third equation shows that the total amount of interest earned from each fund equals $670.<\/p>\n<p style=\"text-align: center;\">[latex]0.03x+0.04y+0.07z=670[\/latex]<\/p>\n<p>Then, we write the three equations as a system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 12,000 \\\\ -y+z &=& 4,000 \\\\ 0.03x+0.04y+0.07z &=& 670 \\end{array}[\/latex]<\/p>\n<p>To make the calculations simpler, we can multiply the third equation by 100. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 12,000 && (1) \\\\ -y+z &=& 4,000 && (2) \\\\ 3x+4y+7z &=& 67,000 && (3) \\end{array}[\/latex]<\/p>\n<p>Step 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up.<\/p>\n<p style=\"text-align: center;\">[latex][\/latex]<\/p>\n<p>Step 2. Multiply equation (1) by [latex]-3[\/latex] and add to equation (2). Write the result as row 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 12,000 \\\\ y+4z &=& 31,000 \\\\ -y+z &=& 4,000 \\end{array}[\/latex]<\/p>\n<p>Step 3. Add equation (2) to equation (3) and write the result as equation (3).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 12,000 \\\\ y+4z &=& 31,000 \\\\ 5z &=& 35,000 \\end{array}[\/latex]<\/p>\n<p>Step 4. Solve for <em>z<\/em> in equation (3). Back-substitute that value in equation (2) and solve for <em>y<\/em>. Then, back-substitute the values for <em>z<\/em> and <em>y<\/em> into equation (1) and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 5z &=& 35,000 \\\\ z &=& 7,000 \\\\ \\\\ \\\\ y+4(7,000) &=& 31,000 \\\\ y &=& 3,000 \\\\ \\\\ \\\\ x+3,000+7,000 &=& 12,000 \\\\ x &=& 2,000 \\end{array}[\/latex]<\/p>\n<p>Jordi invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds.<\/p>\n<\/details>\n<\/div>\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 system of equations in three variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y-2z &=& -1\\\\ 3x-3y-z &=& 5 \\\\ x-2y+3z &=& 6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134162165\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Three Variables<\/h2>\n<p id=\"fs-id1165135541808\">Just as with systems of equations in two variables, we may come across an <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">inconsistent system<\/span> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as [latex]3=7[\/latex] or some other contradiction.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Solving an Inconsistent System of Three Equations in Three Variables<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} cx-3y+z &=& 4 && (1) \\\\ -x+2y-5z &=& 3 && (2) \\\\ 5x-13y+13z &=& 8 && (3) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Looking at the coefficients of <em>x<\/em> we can see that we can eliminate <em>x<\/em> by adding equation (1) to equation (2).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-3y+z &=& 4 && (1) \\\\ -x+2y-5z &=& 3 && (2) \\\\ \\hline -y-4z &=& 7 && (4) \\end{array}[\/latex]<\/p>\n<p>Next, we multiply equation (1) by [latex]-5[\/latex] and add it to equation (3).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -5x+15y-5z &=& -20 && (1) \\ \\text{multiplied by } -5 \\\\ 5x-13y+13z &-& 8 && (3) \\\\ \\hline 2y+8z &=& -12 && (5) \\end{array}[\/latex]<\/p>\n<p>Then, we multiply equation (4) by 2 and add it to equation (5).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -2y-8z &=& 14 && (4) \\ \\text{multiplied by } 2 \\\\ 2y+8z &=& -12 && (5) \\\\ \\hline 0 &=& 2 \\end{array}[\/latex]<\/p>\n<p>The final equation [latex]0=2[\/latex] is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.<\/p>\n<h3>Analysis<\/h3>\n<p>In this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165135315557\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137678319\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system of three equations in three variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 2 \\\\ y-3z &=& 1 \\\\ 2x+y+5z &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134279181\" data-depth=\"1\">\n<h2 data-type=\"title\">Expressing the Solution of a System of Dependent Equations Containing Three Variables<\/h2>\n<p id=\"fs-id1165134279186\">We know from working with systems of equations in two variables that a <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">dependent system<\/span> of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding the Solution to a Dependent System of Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the solution to the given system of three equations in three variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y-3z &=& 0 && (1) \\\\ 4x+2y-6z &=& 0 && (2) \\\\ x-y+z &=& 0 && (3) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, we can multiply equation (1) by [latex]-2[\/latex] and add it to equation (2).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -4x-2y+6z &=& 0 && \\text{equation (1) multiplied by } -2 \\\\ 4x+2y-6z &=& 0 && (2) \\\\ \\hline 0=0 \\end{array}[\/latex]<\/p>\n<p>We do not need to proceed any further. The result we get is an identity, [latex]0=0[\/latex] which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by [latex]-2[\/latex] and adding it to equation (1). We then perform the same steps as above and find the same result, [latex]0=0.[\/latex]<\/p>\n<p>When a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y-3z &=& 0 \\\\ x-y+z &=& 0 \\\\ \\hline 3x-2z &=& 0 \\end{array}[\/latex]<\/p>\n<p>We then solve the resulting equation for <em>z<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3x-2z &=& 0 \\\\ z &=& \\frac{3}{2}x \\end{array}[\/latex]<\/p>\n<p>We back-substitute the expression for <em>z<\/em> into one of the equations and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y-3\\left(\\frac{3}{2}x\\right) &=& 0 \\\\ 2x+y-\\frac{9}{2}x &=& 0 \\\\ y &=& \\frac{9}{2}x-2x \\\\ y &=& \\frac{5}{2}x \\end{array}[\/latex]<\/p>\n<p>So the general solution is [latex]\\left(x, \\frac{5}{2}x, \\frac{3}{2}x\\right).[\/latex] In this solution, <em>x<\/em> can be any real number. The values of <em>y<\/em> and <em>z<\/em> are dependent on the value selected for <em>x<\/em>.<\/p>\n<h3>Analysis<\/h3>\n<p>As shown in Figure 5, two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.<\/p>\n<figure id=\"attachment_1048\" aria-describedby=\"caption-attachment-1048\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1048\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-300x238.webp\" alt=\"\" width=\"300\" height=\"238\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-300x238.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-65x51.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-225x178.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5-350x277.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.2-fig-5.webp 437w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1048\" class=\"wp-caption-text\">Figure 5<\/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>Q: Does the generic solution to a dependent system always have to be written in terms of x? <\/strong><\/p>\n<p><em data-effect=\"italics\">A: No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed x and y. <\/em><\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\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>Solve the following system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+y+z &=& 7 \\\\ 3x-2y-z &=& 4 \\\\ x+6y+5z &=& 24 \\end{array}[\/latex]<\/p>\n<\/div>\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 systems of equations in three variables.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=3RbVSvvRyeI\">Ex 1: System of Three Equations with Three Unknowns Using Elimination<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=r6htz3gaHZ0\">Ex 2: System of Three Equations with Three Unknowns Using Elimination<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_09_02_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135191971\" data-type=\"problem\"><\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.2 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137428022\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137581426\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137581432\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135582116\" 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-id1165137581432-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can a linear system of three equations have exactly two solutions? Explain why or why not<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134116743\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134116745\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135421454\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135421456\" 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-id1165135421454-solution\">3<\/a><span class=\"os-divider\">. <\/span>If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135354941\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135354943\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Using the method of addition, is there only one way to solve the system?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137611460\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133238406\" 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-id1165137611460-solution\">5<\/a><span class=\"os-divider\">. <\/span>Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134205883\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165135161479\">For the following exercises, determine whether the ordered triple given is the solution to the system of equations.<\/p>\n<div id=\"fs-id1165135161483\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135432898\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x-6y+6z &=& -12 \\\\ x+4y+5z &=& -1 \\\\ -x+2y+3z &=& -1 \\end{array}[\/latex] and [latex](0, 1, -1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132079296\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132079298\" 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-id1165132079296-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 6x-y+3z&=& 6 \\\\ 3x+5y+2z &\\& 0 \\\\ x+y &=& 0 \\end{array}[\/latex] and [latex](3, -3, -5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134430325\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134430327\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 6x-7y+z &=& 2 \\\\ -x-y+3z &+& 4 \\\\ 2x+y-z &=& 1 \\end{array}[\/latex] and [latex](4, 2,-6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133306920\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133306922\" 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-id1165133306920-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x-y &=& 0 \\\\ x-z &=& 5 \\\\ x-y+z &=& -1 \\end{array}[\/latex] and [latex](4, 4, -1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135577081\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135553486\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x-y+2z &=& 3 \\\\ 5x+8y-3z &=& 4 \\\\ -x+3y-5z &=& -5 \\end{array}[\/latex] and [latex](4, 1, -7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137823121\">For the following exercises, solve each system by elimination.<\/p>\n<div id=\"fs-id1165137823124\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137823126\" 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-id1165137823124-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-4y+2z &=& -15 \\\\ 2x+4y+z &=& 16 \\\\ 2x+3y+5z &=& 20 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137784201\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137784203\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-2y+3z &=& 20 \\\\ 2x-4y-3z &=& -9 \\\\ x+6y-8z &=& 21 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137898682\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898684\" 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-id1165137898682-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x+2y+4z &=& 9 \\\\ -3x+2y+z &=& 10 \\\\ 4x-3y+5z &=& -3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135609190\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135609192\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 4x-3y+5z &=& 31 \\\\ -x+2y+4z &=& 20 \\\\ x+5y-2z &=& -29 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134089381\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134089383\" 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-id1165134089381-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-2y+3z &=& 4 \\\\ -4x+6y-7z &=& -1 \\\\ 3x+2y-z &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135702605\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135702607\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 4x+6y+9z &=& 0 \\\\ -5x+2y-6z &=& 3 \\\\ 7x-4y+3z &=& -3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134280366\">For the following exercises, solve each system by Gaussian elimination.<\/p>\n<div id=\"fs-id1165133360321\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133360323\" 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-id1165133360321-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x-y+3z &=& 17 \\\\ -5x+4y-2z &=& -46 \\\\ 2y+5z&=& -7 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133046785\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133046787\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-6y+3z &=& 50 \\\\ -x+4y &=& 10 \\\\ 2x-z &=& 10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131907322\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131907324\" 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-id1165131907322-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x+3y-6z &=& 1 \\\\ -4x=6y+12z &=& -2 \\\\ x+2y+5z &=& 10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135389856\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135389858\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 4x+6y-2z &=& 8 \\\\ 6x+9y-3z &=& 12 \\\\ -2x-3y+z &=& -4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135633857\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134216211\" 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-id1165135633857-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x+3y-4z &=& 5 \\\\ -3x+2y+z &=& 11 \\\\ -x+5y+3z &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135707939\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135707941\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 10x+2y-14z &=& 8 \\\\ -x-2y-4z &=& -1 \\\\ -12x-6y+6z &=& -12 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134368156\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134368158\" 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-id1165134368156-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y+z &=& 14 \\\\ 2y+3z &=& -14 \\\\ -16y-24z &=& -112 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134072235\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135620859\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-3y+4z &=& -1 \\\\ -4x+2y-3z &=& 0 \\\\ -x+5y+7z &=& -11 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137938397\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137938399\" 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-id1165137938397-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y+z &=& 0 \\\\ 2x-y+3z &=& 0 \\\\ x-z &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135602254\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135602256\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x+2y-5z &=& 6 \\\\ 5x-4y+3z &=& -12 \\\\ 4x+5y-2z &=& 15 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134037555\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134037557\" 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-id1165134037555-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y+z &=& 0 \\\\ 2x-y+3z &=& 0 \\\\ x-z &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133391507\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133391509\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-\\frac{1}{2}y-z &=& -\\frac{1}{2} \\\\ 4x+z &=& 3 \\\\ -x+\\frac{3}{2}y &=& \\frac{5}{2} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135316107\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135316109\" 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-id1165135316107-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 6x-5y+6z &=& 38 \\\\ \\frac{1}{5}x-\\frac{1}{2}y+\\frac{3}{5}z &=& 1 \\\\ -4x-\\frac{3}{2}y-z &=& -74 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135516682\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135516684\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{2}x-\\frac{1}{5}y+\\frac{2}{5}z &=& -\\frac{13}{10} \\\\ \\frac{1}{4}x-\\frac{2}{5}y-\\frac{1}{5}z &=& -\\frac{7}{20} \\\\ -\\frac{1}{2}x-\\frac{3}{4}y-\\frac{1}{2}z &=& -\\frac{5}{4} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133358779\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135253693\" 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-id1165133358779-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -\\frac{1}{3}x-\\frac{1}{2}y-\\frac{1}{4}z &=& \\frac{3}{4} \\\\ -\\frac{1}{2}x-\\frac{1}{4}y-\\frac{1}{2}z &=& 2 \\\\ -\\frac{1}{4}x-\\frac{3}{4}y-\\frac{1}{2}z &=& -\\frac{1}{2} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132005229\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132005232\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{2}x-\\frac{1}{4}y+\\frac{3}{4}z &=& 0 \\\\ \\frac{1}{4}x+\\frac{1}{10}y+\\frac{2}{5}z &=& -2 \\\\ \\frac{1}{8}x+\\frac{1}{5}y-\\frac{1}{8}z &=& 2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199541\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199543\" 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-id1165134199541-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{4}{5}x-\\frac{7}{8}y+\\frac{1}{2}z &=& 1 \\\\ -\\frac{4}{5}x-\\frac{3}{4}y+\\frac{1}{3}z &=& -8 \\\\ -\\frac{2}{5}x-\\frac{7}{8}y+\\frac{1}{2}z &=& -5 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137896940\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137896942\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -\\frac{1}{3}x-\\frac{1}{8}y+\\frac{1}{6}z &=& -\\frac{4}{3} \\\\ -\\frac{2}{3}x-\\frac{7}{8}y+\\frac{1}{3}z &=& -\\frac{23}{3} \\\\ -\\frac{1}{3}x-\\frac{5}{8}y+\\frac{5}{6}z &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135485720\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485722\" 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-id1165135485720-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -\\frac{1}{4}x-\\frac{5}{4}y+\\frac{5}{2}z &=& -5 \\\\ -\\frac{1}{2}x-\\frac{5}{2}y+\\frac{5}{4}z &=& \\frac{55}{12} \\\\ -\\frac{1}{3}x-\\frac{1}{3}y+\\frac{1}{3}z &=& \\frac{5}{3} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134254135\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134254137\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{40}x+\\frac{1}{60}y+\\frac{1}{80}z &=& \\frac{1}{100} \\\\ -\\frac{1}{2}x-\\frac{1}{3}y-\\frac{1}{4}z &=& -\\frac{1}{5} \\\\ \\frac{3}{8}x+\\frac{3}{12}y+\\frac{3}{16}z &=& \\frac{3}{20} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135417018\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135417020\" 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-id1165135417018-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.1x-0.2y+0.3z &=& 2 \\\\ 0.5x=0.1y+0.4z &=& 8 \\\\ 0.7x-0.2y+0.3z &=& 8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134277203\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134277205\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.2x+0.1y-0.3z &=& 0.2 \\\\ 0.8x+0.4y-1.2z &=& 0.1 \\\\ 1.6x+0.8y-2.4z &=& 0.2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135450286\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135450288\" 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-id1165135450286-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 1.1x+0.7y-3.1z &=& -1.79 \\\\ 2.1x+0.5y-1.6z &=& -0.13 \\\\ 0.5x+0.4y-0.5z &=& -0.07 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135202434\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133134690\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.5x-0.5+0.5z &=& 10 \\\\ 0.2x-0.2y+0.2z &=& 4 \\\\ 0.1x-0.1y+0.1z &=& 2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133194612\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133194614\" 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-id1165133194612-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.1x+0.2y+0.3z &=& 0.37 \\\\ 0.1x-0.2y-0.3z &=& -0.27 \\\\ 0.5x-0.1y-0.3z &=& -0.03 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135672780\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135672782\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.5x-0.5y-0.3z &=& 0.13 \\\\ 0.4x-0.1y-0.3z &=& 0.11 \\\\ 0.2x-0.8y-0.9z &=& -0.32 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135640563\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135640565\" 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-id1165135640563-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.5x+0.2y-0.3z &=& 1 \\\\ 0.4x-0.6y+0.7z &=& 0.8 \\\\ 0.3x-0.1y-0.9z &=& 0.6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133393400\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133393403\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.3x+0.3y+0.5x &=& 0.6 \\\\ 0.4x+0.4y+0.4z &=& 1.8 \\\\ 0.4x+0.2y+0.1z &=& 1.6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132005156\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132005158\" 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-id1165132005156-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.8x+0.8y+0.8z &=& 2.4 \\\\ 0.3x-0.5y+0.2z &=& 0 \\\\ 0.1x+0.2y+0.3z &=& 0.6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165133294287\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165135537327\">For the following exercises, solve the system for x, y, and z.<\/p>\n<div id=\"fs-id1165133441059\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133441061\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y+z &=& 3 \\\\ \\frac{x-1}{2}+\\frac{y-3}{2}+\\frac{z+1}{2} &=& 0 \\\\ \\frac{x-2}{3}+\\frac{y+4}{3}+\\frac{z-3}{3} &=& \\frac{2}{3} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135539006\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135539008\" 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-id1165135539006-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-3y-\\frac{z+1}{2} &=& \\frac{1}{2} \\\\ 6x+\\frac{y-9}{2}+2z &=& -3 \\\\ \\frac{x+8}{2}-4y+z &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133320194\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133320196\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{x+4}{7}-\\frac{y-1}{6}+\\frac{z+2}{3} &=& 1 \\\\ \\frac{x-2}{4}+\\frac{y+1}{8}-\\frac{z+8}{12} &=& 0 \\\\ \\frac{x+6}{3}-\\frac{y+2}{3}+\\frac{z+4}{2} &=& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132035881\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132035883\" 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-id1165132035881-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{x-3}{6}+\\frac{y+2}{2}-\\frac{z-3}{3} &=& 2 \\\\ \\frac{x+2}{4}+\\frac{y-5}{2}+\\frac{z+4}{2} &=& 1 \\\\ \\frac{x+6}{2}-\\frac{y-3}{2}+z+1 &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135344036\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135344039\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{x-1}{3}+\\frac{y+3}{4}+\\frac{z+2}{6} &=& 1 \\\\ 4x+3y-2z &=& 11 \\\\ 0.02x+0.015y-0.01z &=& 0.065 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135258841\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165134053995\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134053997\" 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-id1165134053995-solution\">51<\/a><span class=\"os-divider\">. <\/span>Three even numbers sum up to 108. The smaller is half the larger and the middle number is [latex]\\frac{3}{4}[\/latex] the larger. What are the three numbers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134190578\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134190580\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132972738\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132972740\" 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-id1165132972738-solution\">53<\/a><span class=\"os-divider\">. <\/span>At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132972757\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132972759\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>The Aurora animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133309239\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133309241\" 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-id1165133309239-solution\">55<\/a><span class=\"os-divider\">. <\/span>Your roommate, Shani, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate\u2019s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134072281\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134072283\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Your roommate, John, offered to buy household supplies for you and your other roommate. You live near the border of three states, each of which has a different sales tax. The total amount of money spent was $100.75. Your supplies were bought with 5% tax, John\u2019s with 8% tax, and your third roommate\u2019s with 9% sales tax. The total amount of money spent without taxes is $93.50. If your supplies before tax were $1 more than half of what your third roommate\u2019s supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134072299\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137394065\" 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-id1165134072299-solution\">57<\/a><span class=\"os-divider\">. <\/span>Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137394083\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137394085\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>At a carnival at the Aurora Mall, $2,914.25 in receipts were taken at the end of the day. The cost of a child\u2019s ticket was $20.50, an adult ticket was $29.75, and a senior citizen ticket was $15.25. There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135705887\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135705889\" 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-id1165135705887-solution\">59<\/a><span class=\"os-divider\">. <\/span>Skool Daze, an Aurora, CO, band, sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134231621\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134231623\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>In a bag, a child has 325 coins worth $19.50. There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134231632\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134231635\" 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-id1165134231632-solution\">61<\/a><span class=\"os-divider\">. <\/span>Last year, at Stevinson Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $140,000. This year, due to inflation, the same cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year\u2019s Jeep was $7,000 less than the price of last year\u2019s BMW, what was the price of each of the three cars last year?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131950632\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131950634\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>When his youngest child moved out, Deandre sold his home and made three investments using gains from the sale. He invested $80,500 into three accounts, one that paid 4% simple interest, one that paid [latex]3 \\frac{1}{8}\\%[\/latex] simple interest, and one that paid [latex]2 \\frac{1}{2}\\%[\/latex] simple interest. He earned $2,670 interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134138635\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134138637\" 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-id1165134138635-solution\">63<\/a><span class=\"os-divider\">. <\/span>You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134554932\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134554934\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>An entrepreneur sells a portion of their business for one hundred thousand dollars and invests it all in three accounts for one year. The first account pays 4% compounded annually, the second account pays 3% compounded annually, and the third account pays 2% compounded annually. After one year, the entrepreneur earns $3,650 in interest. If they invested five times the money in the account that pays 4% compared to 3%, how much did they invest in each account?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134381545\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134381547\" 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-id1165134381545-solution\">65<\/a><span class=\"os-divider\">. <\/span>The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world\u2019s consumed oil. The United States consumed 0.7% more than four times China\u2019s consumption. The United States consumed 5% more than triple Japan\u2019s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137404768\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137404770\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced 31.4% of the world\u2019s produced oil. Saudi Arabia and the United States combined for 22.1% of the world\u2019s production, and Saudi Arabia produced 2% more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134226686\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134226688\" 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-id1165134226686-solution\">67<\/a><span class=\"os-divider\">. <\/span>The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported 1.8% more from Saudi Arabia than they did from Mexico, and 1.7% more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135367795\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135367797\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for 64% of the United States oil production. The Gulf of Mexico and Texas combined for 47% of oil production. Texas produced 3% more than Alaska. What percent of United States oil production came from these regions?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135367810\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135367812\" 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-id1165135367810-solution\">69<\/a><span class=\"os-divider\">. <\/span>At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134116842\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134116844\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Meat consumption in the United States can be broken into three categories: red meat, poultry, and fish. If fish makes up 4% less than one-quarter of poultry consumption, and red meat consumption is 18.2% higher than poultry consumption, what are the percentages of meat consumption?<\/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-213","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/213","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":15,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/213\/revisions"}],"predecessor-version":[{"id":1698,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/213\/revisions\/1698"}],"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\/213\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=213"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=213"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=213"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}