{"id":2094,"date":"2016-07-02T00:34:37","date_gmt":"2016-07-02T00:34:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=2094"},"modified":"2019-08-06T18:07:47","modified_gmt":"2019-08-06T18:07:47","slug":"solving-systems-of-three-equations-in-three-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/solving-systems-of-three-equations-in-three-variables\/","title":{"raw":"Systems of Three Equations in Three Variables","rendered":"Systems of Three Equations in Three Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve systems of three equations in three variables<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe <strong>solution set<\/strong> to a system of three equations in three variables is an ordered triple [latex]\\left(x,y,z\\right)[\/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.\r\n<div class=\"textbox\">\r\n<h3>Solution Set, One Solution<\/h3>\r\nThe figure below\u00a0illustrates how a\u00a0system with three variables\u00a0can have one solution.\u00a0Systems that have a single solution are those which result in a <strong>solution set<\/strong> consisting of an ordered triple [latex]\\left(x,y,z\\right)[\/latex]. Graphically, the ordered triple defines a point that is the intersection of three planes in space.\r\n\r\n<img class=\"size-full wp-image-2389 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11211012\/Screen-Shot-2016-07-11-at-2.08.59-PM.png\" alt=\"Screen Shot 2016-07-11 at 2.08.59 PM\" width=\"262\" height=\"210\" \/>\r\n\r\n<\/div>\r\nIn the first example, we will determine whether an ordered triple is a solution for a system of three linear equations in three variables.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.\r\n<div>[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"642719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"642719\"]\r\n\r\nWe will check each equation by substituting in the values of the ordered triple for [latex]x,y[\/latex], and [latex]z[\/latex].\r\n\r\n[latex]\\begin{array}{ccccc}\\begin{array}{r}\\hfill x+y+z=2\\\\ \\hfill \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\hfill \\text{True}\\end{array}&amp; &amp; \\begin{array}{r}\\hfill \\text{}6x - 4y+5z=31\\\\ \\hfill 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ \\hfill 18+8+5=31\\\\ \\hfill \\text{True}\\end{array}&amp; &amp; \\begin{array}{r}\\hfill \\text{}5x+2y+2z=13\\\\ \\hfill 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ \\hfill \\text{}15 - 4+2=13\\\\ \\hfill \\text{True}\\end{array}\\end{array}[\/latex]\r\n\r\nThe ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a linear system of three equations, solve for three unknowns<strong>\r\n<\/strong><\/h3>\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\nSolving a system with three variables is very similar to solving one with two variables. It is important to keep track of your work as the addition of one more equation can create more steps in the solution process.\r\n\r\nIn the example that follows, we will solve the system by first using the elimination method to solve for x and then using back-substitution.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the given system.\r\n\r\n[latex]\\displaystyle\\begin{cases}x-\\dfrac{1}{3}y+\\dfrac{1}{2}z=1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y-\\dfrac{1}{2}z=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=-1\\end{cases}[\/latex]\r\n[reveal-answer q=\"538379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"538379\"]\r\n\r\nThe third equation states that\u00a0[latex]z = \u22121[\/latex], so\u00a0we substitute this into the second equation to obtain a solution for\u00a0[latex]y[\/latex].\r\n\r\n[latex]\\begin{array}{l}y-\\dfrac{1}{2}(-1)=4\\\\y+\\dfrac{1}{2}=4\\\\y=4-\\dfrac{1}{2}\\\\y=\\dfrac{8}{2}-\\dfrac{1}{2}\\\\y=\\dfrac{7}{2}\\end{array}[\/latex]\r\n\r\nNow we have two of our solutions, and we can substitute them both into the first equation to solve for\u00a0[latex]x[\/latex].\r\n\r\n[latex]\\begin{array}{l}x-\\dfrac{1}{3}\\left(\\dfrac{7}{2}\\right)+\\dfrac{1}{2}\\left(-1\\right)=1\\\\x-\\dfrac{7}{6}-\\dfrac{1}{2}=1\\\\x-\\dfrac{7}{6}-\\dfrac{3}{6}=1\\\\x-\\dfrac{10}{6}=1\\\\x=1+\\dfrac{10}{6}\\\\x=\\dfrac{6}{6}+\\dfrac{10}{6}\\\\x=\\dfrac{16}{6}=\\dfrac{8}{3}\\end{array}[\/latex]\r\n\r\nNow we have our ordered triple; remember that where you place the solutions matters!\r\n\r\n[latex](x,y,z)=\\left(\\dfrac{8}{3},\\dfrac{7}{2},-1\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Analysis of the Solution:<\/h3>\r\nEach of the lines in the system above represents a plane (think about a sheet of paper). If you imagine three sheets of notebook paper each representing a portion of these planes, you will start to see the complexities involved in how three such planes can intersect. Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line, so our intersection point is where all three of these lines meet.\r\n\r\n[caption id=\"attachment_2377\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11200619\/Screen-Shot-2016-07-11-at-1.04.41-PM-300x237.png\" alt=\"Three Planes Intersecting.\" width=\"300\" height=\"237\" \/> Three planes intersecting.[\/caption]\r\n\r\nIn the following video, we show another example of using back-substitution to solve a system in three variables.\r\n\r\nhttps:\/\/youtu.be\/HHIjTChrIxE\r\n\r\nIn the next example, we will\u00a0not start with a solution, but will need to use the method of elimination to find our first solution.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind a solution to the following system:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x-y+z=5\\,\\,\\,\\,(1)\\\\-2y+z=6\\,\\,\\,\\,(2)\\\\2y-2z=-12\\,\\,\\,\\,(3)\\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"223787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223787\"]\r\n\r\nWe labeled the equations this time to be able to keep track of things a little more easily. The most obvious first step here is to eliminate [latex]y[\/latex] by adding equations (2) and (3).\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\,\\,\\,\\,\\,\\,\\,\\,\\,-2y+z=6\\,\\,\\,\\,(2)\\\\\\underline{\\,\\,\\,\\,2y-2z=-12}\\,\\,\\,\\,(3)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-z=-6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=6\\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nNow we can substitute the value for\u00a0[latex]z[\/latex] that we obtained into equation [latex](2)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rrr}-2y+(6)=6\\\\-2y=6-6\\\\-2y=0\\\\\\,\\,\\,\\,y=0\\end{array}[\/latex]<\/p>\r\nBe careful here not to get confused with a solution of\u00a0[latex]y = 0[\/latex] and an inconsistent solution. \u00a0It is ok for variables to equal\u00a0[latex]0[\/latex].\r\n\r\nNow we can substitute\u00a0[latex]z = 6[\/latex] and\u00a0[latex]y = 0[\/latex] back into the first equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rrr}x-y+z=5\\\\x-0+6=5\\\\x+6=5\\\\x=5-6\\\\x=-1\\end{array}[\/latex]<\/p>\r\n[latex](x,y,z)=(-1,0,6)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following videos, we show more examples of the algebra you may encounter when solving systems with three variables.\r\n\r\nhttps:\/\/youtu.be\/r6htz3gaHZ0\r\n\r\nhttps:\/\/youtu.be\/3RbVSvvRyeI\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>The solution to a system of linear equations in three variables is an ordered triple of the form [latex](x,y,z)[\/latex].<\/li>\r\n \t<li>Solutions can be verified using substitution and the order of operations.<\/li>\r\n \t<li>Systems of three variables can be solved using the same techniques as we used to solve systems with two variables, including elimination and substitution.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve systems of three equations in three variables<\/li>\n<\/ul>\n<\/div>\n<p>The <strong>solution set<\/strong> to a system of three equations in three variables is an ordered triple [latex]\\left(x,y,z\\right)[\/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\">\n<h3>Solution Set, One Solution<\/h3>\n<p>The figure below\u00a0illustrates how a\u00a0system with three variables\u00a0can have one solution.\u00a0Systems that have a single solution are those which result in a <strong>solution set<\/strong> consisting of an ordered triple [latex]\\left(x,y,z\\right)[\/latex]. Graphically, the ordered triple defines a point that is the intersection of three planes in space.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2389 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11211012\/Screen-Shot-2016-07-11-at-2.08.59-PM.png\" alt=\"Screen Shot 2016-07-11 at 2.08.59 PM\" width=\"262\" height=\"210\" \/><\/p>\n<\/div>\n<p>In the first example, we will determine whether an ordered triple is a solution for a system of three linear equations in three variables.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.<\/p>\n<div>[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q642719\">Show Solution<\/span><\/p>\n<div id=\"q642719\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will check each equation by substituting in the values of the ordered triple for [latex]x,y[\/latex], and [latex]z[\/latex].<\/p>\n<p>[latex]\\begin{array}{ccccc}\\begin{array}{r}\\hfill x+y+z=2\\\\ \\hfill \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\hfill \\text{True}\\end{array}& & \\begin{array}{r}\\hfill \\text{}6x - 4y+5z=31\\\\ \\hfill 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ \\hfill 18+8+5=31\\\\ \\hfill \\text{True}\\end{array}& & \\begin{array}{r}\\hfill \\text{}5x+2y+2z=13\\\\ \\hfill 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ \\hfill \\text{}15 - 4+2=13\\\\ \\hfill \\text{True}\\end{array}\\end{array}[\/latex]<\/p>\n<p>The ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a linear system of three equations, solve for three unknowns<strong><br \/>\n<\/strong><\/h3>\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<p>Solving a system with three variables is very similar to solving one with two variables. It is important to keep track of your work as the addition of one more equation can create more steps in the solution process.<\/p>\n<p>In the example that follows, we will solve the system by first using the elimination method to solve for x and then using back-substitution.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the given system.<\/p>\n<p>[latex]\\displaystyle\\begin{cases}x-\\dfrac{1}{3}y+\\dfrac{1}{2}z=1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y-\\dfrac{1}{2}z=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=-1\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q538379\">Show Solution<\/span><\/p>\n<div id=\"q538379\" class=\"hidden-answer\" style=\"display: none\">\n<p>The third equation states that\u00a0[latex]z = \u22121[\/latex], so\u00a0we substitute this into the second equation to obtain a solution for\u00a0[latex]y[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}y-\\dfrac{1}{2}(-1)=4\\\\y+\\dfrac{1}{2}=4\\\\y=4-\\dfrac{1}{2}\\\\y=\\dfrac{8}{2}-\\dfrac{1}{2}\\\\y=\\dfrac{7}{2}\\end{array}[\/latex]<\/p>\n<p>Now we have two of our solutions, and we can substitute them both into the first equation to solve for\u00a0[latex]x[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}x-\\dfrac{1}{3}\\left(\\dfrac{7}{2}\\right)+\\dfrac{1}{2}\\left(-1\\right)=1\\\\x-\\dfrac{7}{6}-\\dfrac{1}{2}=1\\\\x-\\dfrac{7}{6}-\\dfrac{3}{6}=1\\\\x-\\dfrac{10}{6}=1\\\\x=1+\\dfrac{10}{6}\\\\x=\\dfrac{6}{6}+\\dfrac{10}{6}\\\\x=\\dfrac{16}{6}=\\dfrac{8}{3}\\end{array}[\/latex]<\/p>\n<p>Now we have our ordered triple; remember that where you place the solutions matters!<\/p>\n<p>[latex](x,y,z)=\\left(\\dfrac{8}{3},\\dfrac{7}{2},-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution:<\/h3>\n<p>Each of the lines in the system above represents a plane (think about a sheet of paper). If you imagine three sheets of notebook paper each representing a portion of these planes, you will start to see the complexities involved in how three such planes can intersect. Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line, so our intersection point is where all three of these lines meet.<\/p>\n<div id=\"attachment_2377\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2377\" class=\"size-medium wp-image-2377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11200619\/Screen-Shot-2016-07-11-at-1.04.41-PM-300x237.png\" alt=\"Three Planes Intersecting.\" width=\"300\" height=\"237\" \/><\/p>\n<p id=\"caption-attachment-2377\" class=\"wp-caption-text\">Three planes intersecting.<\/p>\n<\/div>\n<p>In the following video, we show another example of using back-substitution to solve a system in three variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/HHIjTChrIxE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we will\u00a0not start with a solution, but will need to use the method of elimination to find our first solution.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find a solution to the following system:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x-y+z=5\\,\\,\\,\\,(1)\\\\-2y+z=6\\,\\,\\,\\,(2)\\\\2y-2z=-12\\,\\,\\,\\,(3)\\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223787\">Show Solution<\/span><\/p>\n<div id=\"q223787\" class=\"hidden-answer\" style=\"display: none\">\n<p>We labeled the equations this time to be able to keep track of things a little more easily. The most obvious first step here is to eliminate [latex]y[\/latex] by adding equations (2) and (3).<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\,\\,\\,\\,\\,\\,\\,\\,\\,-2y+z=6\\,\\,\\,\\,(2)\\\\\\underline{\\,\\,\\,\\,2y-2z=-12}\\,\\,\\,\\,(3)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-z=-6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=6\\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Now we can substitute the value for\u00a0[latex]z[\/latex] that we obtained into equation [latex](2)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rrr}-2y+(6)=6\\\\-2y=6-6\\\\-2y=0\\\\\\,\\,\\,\\,y=0\\end{array}[\/latex]<\/p>\n<p>Be careful here not to get confused with a solution of\u00a0[latex]y = 0[\/latex] and an inconsistent solution. \u00a0It is ok for variables to equal\u00a0[latex]0[\/latex].<\/p>\n<p>Now we can substitute\u00a0[latex]z = 6[\/latex] and\u00a0[latex]y = 0[\/latex] back into the first equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rrr}x-y+z=5\\\\x-0+6=5\\\\x+6=5\\\\x=5-6\\\\x=-1\\end{array}[\/latex]<\/p>\n<p>[latex](x,y,z)=(-1,0,6)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following videos, we show more examples of the algebra you may encounter when solving systems with three variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 2: System of Three Equations with Three Unknowns Using Elimination\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/r6htz3gaHZ0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1: System of Three Equations with Three Unknowns Using Elimination\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3RbVSvvRyeI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>The solution to a system of linear equations in three variables is an ordered triple of the form [latex](x,y,z)[\/latex].<\/li>\n<li>Solutions can be verified using substitution and the order of operations.<\/li>\n<li>Systems of three variables can be solved using the same techniques as we used to solve systems with two variables, including elimination and substitution.<\/li>\n<\/ul>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2094\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/Lumen%20Learning\">http:\/\/Lumen%20Learning<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra: 8.1 Systems of Linear Equations: Gaussian Elimination. <strong>Authored by<\/strong>: Stitz, Carl and Zeager, Jeff. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.stitz-zeager.com\/szca07042013.pdf\">http:\/\/www.stitz-zeager.com\/szca07042013.pdf<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/HHIjTChrIxE\">https:\/\/youtu.be\/HHIjTChrIxE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: System of Three Equations with Three Unknowns Using Elimination. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/r6htz3gaHZ0\">https:\/\/youtu.be\/r6htz3gaHZ0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: System of Three Equations with Three Unknowns Using Elimination. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3RbVSvvRyeI\">https:\/\/youtu.be\/3RbVSvvRyeI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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