{"id":773,"date":"2021-09-18T21:32:42","date_gmt":"2021-09-18T21:32:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=773"},"modified":"2021-12-01T19:02:05","modified_gmt":"2021-12-01T19:02:05","slug":"6-2-solving-linear-equations-in-two-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-2-solving-linear-equations-in-two-variables\/","title":{"raw":"6.2: Solving Linear Equations in Two Variables","rendered":"6.2: Solving Linear Equations in Two Variables"},"content":{"raw":"<!-- Le HTML5 shim, for IE6-8 support of HTML5 elements --><!-- [if lt IE 9]>\r\n<script src=\"https:\/\/html5shim.googlecode.com\/svn\/trunk\/html5.js\">\r\n\t<\/script>\r\n<![endif]-->\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Determine whether a given ordered pair is a solution of a given linear equation.<\/li>\r\n \t<li>Find solutions of a linear equation.<\/li>\r\n \t<li>Complete a table of solutions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Ordered pair solution<\/strong>: a solution written in the form [latex]\\left (x,y\\right )[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Finding Solutions of Linear Equations in Two Variables<\/h2>\r\nWhen an equation has two variables, any solution will be an ordered pair with a value for each variable.\r\n<div id=\"post-378\" class=\"standard post-378 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>Solution to a Linear Equation in Two Variables<\/h3>\r\nAn ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution of the linear equation [latex]ax+by=c[\/latex], if the equation is a true statement when the [latex]x[\/latex]- and [latex]y[\/latex]-values of the ordered pair are substituted into the equation.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDetermine whether [latex](\u22122,4)[\/latex] is a solution of the equation [latex]4y+5x=3[\/latex].\r\n<h4>Solution<\/h4>\r\nSubstitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\r\nEvaluate.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\r\nThe statement is not true, so [latex](\u22122,4)[\/latex] is not a solution.\r\n<h4>Answer<\/h4>\r\n[latex](\u22122,4)[\/latex] is not a solution of the equation [latex]4y+5x=3[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine which ordered pairs are solutions of the equation [latex]x+4y=8\\text{:}[\/latex]\r\n\r\n1. [latex]\\left(0,2\\right)[\/latex]\r\n\r\n2. [latex]\\left(2,-4\\right)[\/latex]\r\n\r\n3. [latex]\\left(-4,3\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nSubstitute the [latex]x\\text{- and }y\\text{-values}[\/latex] from each ordered pair into the equation and determine if the result is a true statement.\r\n<table id=\"eip-id1168469838906\" class=\"unnumbered unstyled\" summary=\"This image shows three columns. The first column is labeled \">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<td>2. [latex]\\left(2,-4\\right)[\/latex]<\/td>\r\n<td>3. [latex]\\left(-4,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{2}[\/latex][latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{0}+4\\cdot\\color{red}{2}\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]0+8\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]8=8\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{2}, y=\\color{red}{-4}[\/latex][latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{2}+4(\\color{red}{-4})\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]2+(-16)\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]-14\\not=8[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-4}, y=\\color{red}{3}[\/latex][latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{-4}+4\\cdot\\color{red}{3}\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]-4+12\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]8=8\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,2\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(2,-4\\right)[\/latex] is not a solution.<\/td>\r\n<td>[latex]\\left(-4,3\\right)[\/latex] is a solution.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm146928\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146928&amp;theme=oea&amp;iframe_resize_id=ohm146928&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine which ordered pairs are solutions of the equation. [latex]y=5x - 1\\text{:}[\/latex]\r\n\r\n1. [latex]\\left(0,-1\\right)[\/latex]\r\n\r\n2. [latex]\\left(1,4\\right)[\/latex]\r\n\r\n3. [latex]\\left(-2,-7\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nSubstitute the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] from each ordered pair into the equation and determine if it results in a true statement.\r\n<table id=\"eip-id1168466112848\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions in three columns, a, b, and c. The equation is y equals 5 times x minus 1. The first substitution is for ordered pair (0, -1). The first line in column a reads x = 0, with 0 shown in blue and y = -1, with -1 shown in red. The next line is y = 5 x - 1. The next line is -1, shown in red = 5 open parentheses 0, shown in blue, closed parentheses - 1, with a question mark shown over the equal sign. The next line is -1 = 0 - 1, with a question mark over the equal sign. The next line is - 1 = -1, followed by a check mark. The last line is \">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<td>2. [latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<td>3. [latex]\\left(-2,-7\\right)[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{-1}[\/latex][latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{-1}\\stackrel{?}{=}5(\\color{blue}{0})-1[\/latex]\r\n\r\n[latex]-1\\stackrel{?}{=}0-1[\/latex]\r\n\r\n[latex]-1=-1\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{1}, y=\\color{red}{4}[\/latex][latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{4}\\stackrel{?}{=}5(\\color{blue}{1})-1[\/latex]\r\n\r\n[latex]4\\stackrel{?}{=}5-1[\/latex]\r\n\r\n[latex]4=4\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-2}, y=\\color{red}{-7}[\/latex][latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{-7}\\stackrel{?}{=}5(\\color{blue}{-2})-1[\/latex]\r\n\r\n[latex]-7\\stackrel{?}{=}-10-1[\/latex]\r\n\r\n[latex]-7\\not=-11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,-1\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(1,4\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(-2,-7\\right)[\/latex] is not a solution.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm146929\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146929&amp;theme=oea&amp;iframe_resize_id=ohm146929&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<iframe id=\"ohm146941\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146941&amp;theme=oea&amp;iframe_resize_id=ohm146941&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\nThe video shows more\u00a0examples of how to determine whether an ordered pair is a solution of a linear equation.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/9aWGxt7OnB8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Complete a Table of Solutions<\/h2>\r\nIn the previous examples, we substituted the [latex]x\\text{- and }y\\text{-values}[\/latex] of a given ordered pair to determine whether or not it was a solution of a given linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[\/latex] and then solve the equation for [latex]y[\/latex]. Or, choose a value for [latex]y[\/latex] and then solve for [latex]x[\/latex].\r\n\r\nLet's consider the equation [latex]y=5x - 1[\/latex]. The easiest value to choose for [latex]x[\/latex] or [latex]y[\/latex] is zero:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}y &amp; =5x-1 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Substitute}\\;x=0\\\\y &amp; = 5(0)-1\\\\y &amp; = -1\\end{aligned}\\end{equation}[\/latex]\u00a0 \u00a0 \u00a0 So, [latex]x=0,\\;y=-1[\/latex] is a solution, which as an ordered pair is [latex]\\left (0,\\,-1\\right )[\/latex].\r\n\r\n[latex]\\begin{equation}\\begin{aligned}y &amp; =5x-1 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Substitute}\\;y=0\\\\0 &amp; = 5x-1\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Solve for}\\;x\\\\1 &amp; = 5x\\\\ \\frac{1}{5} &amp; =x\\end{aligned}\\end{equation}[\/latex]\u00a0 \u00a0 \u00a0 So, [latex]x=\\frac{1}{5},\\;y=0[\/latex] is a solution, which as an ordered pair is [latex]\\left (\\frac{1}{5},\\,0\\right )[\/latex].\r\n\r\nWe can continue to find more solutions by choosing different values of [latex]x[\/latex] and [latex]y[\/latex].\r\n\r\nSuppose [latex]x=2[\/latex]:\r\n<table id=\"eip-id1168469817063\" class=\"unnumbered\" summary=\"The figure shows a substitution into an equation and accompanying comments. The first equation is y = 5 open parentheses 2, shown in blue, closed parentheses - 1. The comment is \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]y=5x - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]x=2[\/latex]<\/td>\r\n<td>[latex]y=5(\\color{blue}{2})-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]y=10 - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo find a third solution, we\u2019ll let [latex]x=2[\/latex] and solve for [latex]y[\/latex].\r\n\r\nWe can write our solutions in a table:\r\n<table id=\"fs-id1569134\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 5 x - 1. The next row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{1}{5}[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(\\frac{1}{5},0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]\\left(2,9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can find more solutions to the equation by substituting any value of [latex]x[\/latex] or any value of [latex]y[\/latex] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nComplete the table to find three solutions of the equation [latex]y=4x - 2\\text{:}[\/latex]\r\n<table id=\"fs-id1599948\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nSubstitute [latex]x=0,x=-1[\/latex], and [latex]x=2[\/latex] into [latex]y=4x - 2[\/latex].\r\n<table id=\"eip-id1168468326216\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation. The first substitution x = 0, with 0 shown in blue. The next line is y = 4 x - 2. The next line is y = 4 times 0, shown in blue, minus 2. The next line is y = 0 - 2. The next line is y = -2. The last line is \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-1}[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=4\\cdot{\\color{blue}{0}}-2[\/latex]<\/td>\r\n<td>[latex]y=4(\\color{blue}{-1})-2[\/latex]<\/td>\r\n<td>[latex]y=4\\cdot{\\color{blue}{2}}-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=0 - 2[\/latex]<\/td>\r\n<td>[latex]y=-4 - 2[\/latex]<\/td>\r\n<td>[latex]y=8 - 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=-2[\/latex]<\/td>\r\n<td>[latex]y=-6[\/latex]<\/td>\r\n<td>[latex]y=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe results are summarized in the table.\r\n<table id=\"fs-id1572080\" class=\"unnumbered\" summary=\"This table has 5 rows and three columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-6[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm146945\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146945&amp;theme=oea&amp;iframe_resize_id=ohm146945&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<iframe id=\"ohm146947\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146947&amp;theme=oea&amp;iframe_resize_id=ohm146947&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nComplete the table to find three solutions to the equation [latex]5x - 4y=20\\text{:}[\/latex]\r\n<table id=\"fs-id1328205\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table is 5 rows and 3 columns. The first row is the equation 5 x - 4 y = 20. The next row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px;\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px;\"><\/td>\r\n<td style=\"width: 218.75px;\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\"><\/td>\r\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\"><\/td>\r\n<td style=\"width: 114px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224816\/CNX_BMath_Figure_11_01_038_img.png\" alt=\"The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is \" \/>\r\n\r\nThe results are summarized in the table.\r\n<table id=\"fs-id1572845\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table has 5 rows and 3 columns. The first row is equation 5 x - 4 y = 20. The next row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px;\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\">[latex]\\left(4,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 114px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\">[latex]\\left(8,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm146948\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146948&amp;theme=oea&amp;iframe_resize_id=ohm146948&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\nTo find a solution to a linear equation, we can choose any number we want to substitute into the equation for either [latex]x[\/latex] or [latex]y[\/latex]. We could choose [latex]1,100,-1,000, -\\frac{4}{5}, 2.6[\/latex], or any other value we want. But it\u2019s a good idea to choose a number that\u2019s easy to work with. We\u2019ll usually choose [latex]0[\/latex] as one of our values.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind a solution to the equation [latex]3x+2y=6[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168469770411\" class=\"unnumbered unstyled\" summary=\"The figure shows a four step solution. Step 1 reads \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1:<\/strong>\r\nChoose any value for one of the variables in the equation.<\/td>\r\n<td>We can substitute any value we want for [latex]x[\/latex] or any value for [latex]y[\/latex].Let\u2019s pick [latex]x=0[\/latex].\r\n\r\nWhat is the value of [latex]y[\/latex] if [latex]x=0[\/latex] ?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2:<\/strong>\r\nSubstitute that value into the equation.Solve for the other variable.<\/td>\r\n<td>Substitute [latex]0[\/latex] for [latex]x[\/latex].Simplify.\r\n\r\nDivide both sides by [latex]2[\/latex].<\/td>\r\n<td>[latex]3x+2y=6[\/latex][latex]3\\cdot\\color{blue}{0}+2y=6[\/latex]\r\n\r\n[latex]0+2y=6[\/latex]\r\n\r\n[latex]2y=6[\/latex]\r\n\r\n[latex]y=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3:<\/strong>\r\nWrite the solution as an ordered pair.<\/td>\r\n<td>So, when [latex]x=0,y=3[\/latex].<\/td>\r\n<td>This solution is represented by the ordered pair [latex]\\left(0,3\\right)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4:<\/strong>\r\nCheck.<\/td>\r\n<td>Substitute [latex]x=\\color{blue}{0}, y=\\color{red}{3}[\/latex] into the equation [latex]3x+2y=6[\/latex]Is the result a true equation?\r\n\r\nYes!<\/td>\r\n<td>[latex]3x+2y=6[\/latex][latex]3\\cdot\\color{blue}{0}+2\\cdot\\color{red}{3}\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]0+6\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6=6\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm147000\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147000&amp;theme=oea&amp;iframe_resize_id=ohm147000&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm147003\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147003&amp;theme=oea&amp;iframe_resize_id=ohm147003&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind three solutions to the equation [latex]x - 4y=8[\/latex].\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168468771120\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first starts with the equation x - 4 y = 8. The next line is x = 0, with 0 shown in blue. The next line is 0 - 4 y = 8, with 0 shown in blue. The comment is \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choose a value for [latex]x[\/latex] or [latex]y[\/latex].<\/td>\r\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\r\n<td>[latex]y=\\color{red}{0}[\/latex]<\/td>\r\n<td>[latex]y=\\color{red}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute it into the equation.<\/td>\r\n<td>[latex]\\color{blue}{0}-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4\\cdot\\color{red}{0}=8[\/latex]<\/td>\r\n<td>[latex]x-4\\cdot\\color{red}{3}=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Solve.<\/td>\r\n<td>[latex]-4y=8[\/latex][latex]y=-2[\/latex]<\/td>\r\n<td>[latex]x-0=8[\/latex][latex]x=8[\/latex]<\/td>\r\n<td>[latex]x-12=8[\/latex][latex]x=20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the ordered pair.<\/td>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(20,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]\\left(0,-2\\right),\\left(8,0\\right)[\/latex], and [latex]\\left(20,3\\right)[\/latex] are three solutions to the equation [latex]x - 4y=8[\/latex].\r\n<table id=\"fs-id1580614\" class=\"unnumbered\" summary=\"This table it titled x - 4 y =8. It has 4 rows and 3 columns. The first row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]x - 4y=8[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]\\left(20,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRemember, there are an infinite number of solutions to each linear equation. Any ordered pair we find is a solution if it makes the equation true.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"ohm147004\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147004&amp;theme=oea&amp;iframe_resize_id=ohm147004&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p><!-- Le HTML5 shim, for IE6-8 support of HTML5 elements --><!-- [if lt IE 9]>\n<script src=\"https:\/\/html5shim.googlecode.com\/svn\/trunk\/html5.js\">\n\t<\/script>\n<![endif] --><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Determine whether a given ordered pair is a solution of a given linear equation.<\/li>\n<li>Find solutions of a linear equation.<\/li>\n<li>Complete a table of solutions.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Ordered pair solution<\/strong>: a solution written in the form [latex]\\left (x,y\\right )[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Finding Solutions of Linear Equations in Two Variables<\/h2>\n<p>When an equation has two variables, any solution will be an ordered pair with a value for each variable.<\/p>\n<div id=\"post-378\" class=\"standard post-378 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>Solution to a Linear Equation in Two Variables<\/h3>\n<p>An ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution of the linear equation [latex]ax+by=c[\/latex], if the equation is a true statement when the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values of the ordered pair are substituted into the equation.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Determine whether [latex](\u22122,4)[\/latex] is a solution of the equation [latex]4y+5x=3[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\n<p>Evaluate.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\n<p>The statement is not true, so [latex](\u22122,4)[\/latex] is not a solution.<\/p>\n<h4>Answer<\/h4>\n<p>[latex](\u22122,4)[\/latex] is not a solution of the equation [latex]4y+5x=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine which ordered pairs are solutions of the equation [latex]x+4y=8\\text{:}[\/latex]<\/p>\n<p>1. [latex]\\left(0,2\\right)[\/latex]<\/p>\n<p>2. [latex]\\left(2,-4\\right)[\/latex]<\/p>\n<p>3. [latex]\\left(-4,3\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Substitute the [latex]x\\text{- and }y\\text{-values}[\/latex] from each ordered pair into the equation and determine if the result is a true statement.<\/p>\n<table id=\"eip-id1168469838906\" class=\"unnumbered unstyled\" summary=\"This image shows three columns. The first column is labeled\">\n<tbody>\n<tr>\n<td>1. [latex]\\left(0,2\\right)[\/latex]<\/td>\n<td>2. [latex]\\left(2,-4\\right)[\/latex]<\/td>\n<td>3. [latex]\\left(-4,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{2}[\/latex][latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{0}+4\\cdot\\color{red}{2}\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]0+8\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]8=8\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{2}, y=\\color{red}{-4}[\/latex][latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{2}+4(\\color{red}{-4})\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]2+(-16)\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]-14\\not=8[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-4}, y=\\color{red}{3}[\/latex][latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{-4}+4\\cdot\\color{red}{3}\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]-4+12\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]8=8\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,2\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(2,-4\\right)[\/latex] is not a solution.<\/td>\n<td>[latex]\\left(-4,3\\right)[\/latex] is a solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146928\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146928&amp;theme=oea&amp;iframe_resize_id=ohm146928&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine which ordered pairs are solutions of the equation. [latex]y=5x - 1\\text{:}[\/latex]<\/p>\n<p>1. [latex]\\left(0,-1\\right)[\/latex]<\/p>\n<p>2. [latex]\\left(1,4\\right)[\/latex]<\/p>\n<p>3. [latex]\\left(-2,-7\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Substitute the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] from each ordered pair into the equation and determine if it results in a true statement.<\/p>\n<table id=\"eip-id1168466112848\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions in three columns, a, b, and c. The equation is y equals 5 times x minus 1. The first substitution is for ordered pair (0, -1). The first line in column a reads x = 0, with 0 shown in blue and y = -1, with -1 shown in red. The next line is y = 5 x - 1. The next line is -1, shown in red = 5 open parentheses 0, shown in blue, closed parentheses - 1, with a question mark shown over the equal sign. The next line is -1 = 0 - 1, with a question mark over the equal sign. The next line is - 1 = -1, followed by a check mark. The last line is\">\n<tbody>\n<tr>\n<td>1. [latex]\\left(0,-1\\right)[\/latex]<\/td>\n<td>2. [latex]\\left(1,4\\right)[\/latex]<\/td>\n<td>3. [latex]\\left(-2,-7\\right)[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{-1}[\/latex][latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{-1}\\stackrel{?}{=}5(\\color{blue}{0})-1[\/latex]<\/p>\n<p>[latex]-1\\stackrel{?}{=}0-1[\/latex]<\/p>\n<p>[latex]-1=-1\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{1}, y=\\color{red}{4}[\/latex][latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{4}\\stackrel{?}{=}5(\\color{blue}{1})-1[\/latex]<\/p>\n<p>[latex]4\\stackrel{?}{=}5-1[\/latex]<\/p>\n<p>[latex]4=4\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-2}, y=\\color{red}{-7}[\/latex][latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{-7}\\stackrel{?}{=}5(\\color{blue}{-2})-1[\/latex]<\/p>\n<p>[latex]-7\\stackrel{?}{=}-10-1[\/latex]<\/p>\n<p>[latex]-7\\not=-11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,-1\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(1,4\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(-2,-7\\right)[\/latex] is not a solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146929\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146929&amp;theme=oea&amp;iframe_resize_id=ohm146929&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146941\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146941&amp;theme=oea&amp;iframe_resize_id=ohm146941&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The video shows more\u00a0examples of how to determine whether an ordered pair is a solution of a linear equation.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/9aWGxt7OnB8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Complete a Table of Solutions<\/h2>\n<p>In the previous examples, we substituted the [latex]x\\text{- and }y\\text{-values}[\/latex] of a given ordered pair to determine whether or not it was a solution of a given linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[\/latex] and then solve the equation for [latex]y[\/latex]. Or, choose a value for [latex]y[\/latex] and then solve for [latex]x[\/latex].<\/p>\n<p>Let&#8217;s consider the equation [latex]y=5x - 1[\/latex]. The easiest value to choose for [latex]x[\/latex] or [latex]y[\/latex] is zero:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}y & =5x-1 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Substitute}\\;x=0\\\\y & = 5(0)-1\\\\y & = -1\\end{aligned}\\end{equation}[\/latex]\u00a0 \u00a0 \u00a0 So, [latex]x=0,\\;y=-1[\/latex] is a solution, which as an ordered pair is [latex]\\left (0,\\,-1\\right )[\/latex].<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}y & =5x-1 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Substitute}\\;y=0\\\\0 & = 5x-1\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Solve for}\\;x\\\\1 & = 5x\\\\ \\frac{1}{5} & =x\\end{aligned}\\end{equation}[\/latex]\u00a0 \u00a0 \u00a0 So, [latex]x=\\frac{1}{5},\\;y=0[\/latex] is a solution, which as an ordered pair is [latex]\\left (\\frac{1}{5},\\,0\\right )[\/latex].<\/p>\n<p>We can continue to find more solutions by choosing different values of [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p>Suppose [latex]x=2[\/latex]:<\/p>\n<table id=\"eip-id1168469817063\" class=\"unnumbered\" summary=\"The figure shows a substitution into an equation and accompanying comments. The first equation is y = 5 open parentheses 2, shown in blue, closed parentheses - 1. The comment is\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y=5x - 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]x=2[\/latex]<\/td>\n<td>[latex]y=5(\\color{blue}{2})-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]y=10 - 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To find a third solution, we\u2019ll let [latex]x=2[\/latex] and solve for [latex]y[\/latex].<\/p>\n<p>We can write our solutions in a table:<\/p>\n<table id=\"fs-id1569134\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 5 x - 1. The next row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{1}{5}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(\\frac{1}{5},0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]\\left(2,9\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can find more solutions to the equation by substituting any value of [latex]x[\/latex] or any value of [latex]y[\/latex] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Complete the table to find three solutions of the equation [latex]y=4x - 2\\text{:}[\/latex]<\/p>\n<table id=\"fs-id1599948\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]-1[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Substitute [latex]x=0,x=-1[\/latex], and [latex]x=2[\/latex] into [latex]y=4x - 2[\/latex].<\/p>\n<table id=\"eip-id1168468326216\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation. The first substitution x = 0, with 0 shown in blue. The next line is y = 4 x - 2. The next line is y = 4 times 0, shown in blue, minus 2. The next line is y = 0 - 2. The next line is y = -2. The last line is\">\n<tbody>\n<tr>\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-1}[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=4\\cdot{\\color{blue}{0}}-2[\/latex]<\/td>\n<td>[latex]y=4(\\color{blue}{-1})-2[\/latex]<\/td>\n<td>[latex]y=4\\cdot{\\color{blue}{2}}-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=0 - 2[\/latex]<\/td>\n<td>[latex]y=-4 - 2[\/latex]<\/td>\n<td>[latex]y=8 - 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=-2[\/latex]<\/td>\n<td>[latex]y=-6[\/latex]<\/td>\n<td>[latex]y=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The results are summarized in the table.<\/p>\n<table id=\"fs-id1572080\" class=\"unnumbered\" summary=\"This table has 5 rows and three columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-6[\/latex]<\/td>\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146945\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146945&amp;theme=oea&amp;iframe_resize_id=ohm146945&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146947\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146947&amp;theme=oea&amp;iframe_resize_id=ohm146947&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Complete the table to find three solutions to the equation [latex]5x - 4y=20\\text{:}[\/latex]<\/p>\n<table id=\"fs-id1328205\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table is 5 rows and 3 columns. The first row is the equation 5 x - 4 y = 20. The next row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px;\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px;\"><\/td>\n<td style=\"width: 218.75px;\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\"><\/td>\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px;\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\"><\/td>\n<td style=\"width: 114px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224816\/CNX_BMath_Figure_11_01_038_img.png\" alt=\"The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is\" \/><\/p>\n<p>The results are summarized in the table.<\/p>\n<table id=\"fs-id1572845\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table has 5 rows and 3 columns. The first row is equation 5 x - 4 y = 20. The next row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px;\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 218.75px;\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 114px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px;\">[latex]\\left(4,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 114px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px;\">[latex]\\left(8,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146948\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146948&amp;theme=oea&amp;iframe_resize_id=ohm146948&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3><\/h3>\n<p>To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either [latex]x[\/latex] or [latex]y[\/latex]. We could choose [latex]1,100,-1,000, -\\frac{4}{5}, 2.6[\/latex], or any other value we want. But it\u2019s a good idea to choose a number that\u2019s easy to work with. We\u2019ll usually choose [latex]0[\/latex] as one of our values.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find a solution to the equation [latex]3x+2y=6[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168469770411\" class=\"unnumbered unstyled\" summary=\"The figure shows a four step solution. Step 1 reads\">\n<tbody>\n<tr>\n<td><strong>Step 1:<\/strong><br \/>\nChoose any value for one of the variables in the equation.<\/td>\n<td>We can substitute any value we want for [latex]x[\/latex] or any value for [latex]y[\/latex].Let\u2019s pick [latex]x=0[\/latex].<\/p>\n<p>What is the value of [latex]y[\/latex] if [latex]x=0[\/latex] ?<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2:<\/strong><br \/>\nSubstitute that value into the equation.Solve for the other variable.<\/td>\n<td>Substitute [latex]0[\/latex] for [latex]x[\/latex].Simplify.<\/p>\n<p>Divide both sides by [latex]2[\/latex].<\/td>\n<td>[latex]3x+2y=6[\/latex][latex]3\\cdot\\color{blue}{0}+2y=6[\/latex]<\/p>\n<p>[latex]0+2y=6[\/latex]<\/p>\n<p>[latex]2y=6[\/latex]<\/p>\n<p>[latex]y=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3:<\/strong><br \/>\nWrite the solution as an ordered pair.<\/td>\n<td>So, when [latex]x=0,y=3[\/latex].<\/td>\n<td>This solution is represented by the ordered pair [latex]\\left(0,3\\right)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4:<\/strong><br \/>\nCheck.<\/td>\n<td>Substitute [latex]x=\\color{blue}{0}, y=\\color{red}{3}[\/latex] into the equation [latex]3x+2y=6[\/latex]Is the result a true equation?<\/p>\n<p>Yes!<\/td>\n<td>[latex]3x+2y=6[\/latex][latex]3\\cdot\\color{blue}{0}+2\\cdot\\color{red}{3}\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]0+6\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6=6\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147000\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147000&amp;theme=oea&amp;iframe_resize_id=ohm147000&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147003\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147003&amp;theme=oea&amp;iframe_resize_id=ohm147003&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find three solutions to the equation [latex]x - 4y=8[\/latex].<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168468771120\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first starts with the equation x - 4 y = 8. The next line is x = 0, with 0 shown in blue. The next line is 0 - 4 y = 8, with 0 shown in blue. The comment is\">\n<tbody>\n<tr>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Choose a value for [latex]x[\/latex] or [latex]y[\/latex].<\/td>\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\n<td>[latex]y=\\color{red}{0}[\/latex]<\/td>\n<td>[latex]y=\\color{red}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute it into the equation.<\/td>\n<td>[latex]\\color{blue}{0}-4y=8[\/latex]<\/td>\n<td>[latex]x-4\\cdot\\color{red}{0}=8[\/latex]<\/td>\n<td>[latex]x-4\\cdot\\color{red}{3}=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Solve.<\/td>\n<td>[latex]-4y=8[\/latex][latex]y=-2[\/latex]<\/td>\n<td>[latex]x-0=8[\/latex][latex]x=8[\/latex]<\/td>\n<td>[latex]x-12=8[\/latex][latex]x=20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the ordered pair.<\/td>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\n<td>[latex]\\left(20,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]\\left(0,-2\\right),\\left(8,0\\right)[\/latex], and [latex]\\left(20,3\\right)[\/latex] are three solutions to the equation [latex]x - 4y=8[\/latex].<\/p>\n<table id=\"fs-id1580614\" class=\"unnumbered\" summary=\"This table it titled x - 4 y =8. It has 4 rows and 3 columns. The first row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]x - 4y=8[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]\\left(20,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Remember, there are an infinite number of solutions to each linear equation. Any ordered pair we find is a solution if it makes the equation true.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147004\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147004&amp;theme=oea&amp;iframe_resize_id=ohm147004&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\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-773\">\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>Question ID 146941, 146929, 146928, 146927. <strong>Authored by<\/strong>: Lumen Learning. <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>Ex: Determine If An Ordered Pair is a Solution to a Linear Equation. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pJtxugdFjEk\">https:\/\/youtu.be\/pJtxugdFjEk<\/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>Revised and adapted: Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":422608,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 146941, 146929, 146928, 146927\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine If An Ordered Pair is a Solution to a Linear Equation\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pJtxugdFjEk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Revised and adapted: Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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