{"id":1084,"date":"2022-03-12T22:00:30","date_gmt":"2022-03-12T22:00:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1084"},"modified":"2025-11-17T19:53:45","modified_gmt":"2025-11-17T19:53:45","slug":"2-6-1-introduction-to-systems-of-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-6-1-introduction-to-systems-of-linear-equations\/","title":{"raw":"2.6.1: Introduction to Systems of Linear equations in Two Variables","rendered":"2.6.1: Introduction to Systems of Linear equations in Two Variables"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-105\" class=\"standard post-105 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Explain the meaning of a solution or solutions of a system of linear equations<\/li>\r\n \t<li>Explain the three types of systems of linear equations<\/li>\r\n \t<li>Solve a system of linear equations by graphing<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Systems of Equations<\/h2>\r\nA\u00a0<strong><em>system of linear equations\u00a0<\/em><\/strong>consists of two or more linear equations in two or more variables such that all equations in the system are considered simultaneously. To find the single, unique solution of a system of linear equations, we must find a numerical value for each variable in the system that satisfies <em>all<\/em> equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a single, unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a single, unique solution.\r\n\r\nIn this section, we will look at systems of linear equations in two variables, which consist of two equations that each contain the same two variables. For example, consider this system of linear equations in two variables:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}2x+y=\\text{ }15\\\\3x-y=\\text{ }5\\end{array}[\/latex]<\/div>\r\nThe\u00a0<em><strong>solution<\/strong><\/em>\u00a0of a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair [latex](4, 7)[\/latex] is the solution of the system of linear equations. We can verify the solution by substituting the value of each variable into each equation to see if the ordered pair satisfies both equations:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}2\\left(4\\right)+\\left(7\\right)=15\\text{ }\\text{True}\\hfill \\\\ 3\\left(4\\right)-\\left(7\\right)=5\\text{ }\\text{True}\\hfill \\end{array}[\/latex]<\/div>\r\n<span style=\"font-size: 1em;\">This system consists of two linear equations, so graphically, we are considering two lines. Each point on the first line is a solution of the first equation.\u00a0Each point on the second line is a solution of the second equation. Only a point on both lines will satisfy both equations. For the system we just looked at, the two lines cross at the solution (4, 7), as shown in figure 1.<\/span>\r\n\r\n<\/div>\r\n<div class=\"entry-content\">\r\n\r\n[caption id=\"attachment_1477\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1477 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-300x300.png\" alt=\"2 lines crossing at a single point, (4,7)\" width=\"300\" height=\"300\" \/> Figure 1. The intersection point of the two lines is the solution of the system.[\/caption]\r\n\r\n<div id=\"post-105\" class=\"standard post-105 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox\">\r\n<h3>How To determine whether an ordered pair is a solution of a system<\/h3>\r\n<ol>\r\n \t<li>Substitute the ordered pair into each equation in the system.<\/li>\r\n \t<li>Determine whether true statements result from the substitution in each equation; if so, the ordered pair is a solution.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution of the system of equations:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+3y=8\\hfill \\\\ 2x - 9=y\\hfill \\end{array}[\/latex]<\/div>\r\n<h4>Solution<\/h4>\r\nSubstitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into each equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\left(5\\right)+3\\left(1\\right)=8\\hfill &amp; \\hfill \\\\ \\text{ }8=8\\hfill &amp; \\text{True}\\hfill \\\\2\\left(5\\right)-9=\\left(1\\right)\\hfill &amp; \\hfill \\\\ \\text{ }\\text{1=1}\\hfill &amp; \\text{True}\\hfill \\end{array}[\/latex]<\/p>\r\nThe ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution of the system.\r\n\r\nWe can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222634\/CNX_Precalc_Figure_09_01_0032.jpg\" alt=\"A graph of two lines running through the point five, one. The first line's equation is x plus 3y equals 8. The second line's equation is 2x minus 9 equals y.\" width=\"348\" height=\"261\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video shows another example of how to verify whether an ordered pair is a solution of a system of equations.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/2IxgKgjX00k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-1.odt\">Transcript 2.6.1 Video 1<\/a>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine whether the ordered pair [latex]\\left(\u20132,4\\right)[\/latex] is a solution of the system of equations:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}-x+2&amp;=y\\\\2x+y&amp;=8\\end{aligned}[\/latex]<\/p>\r\n\r\n<div><\/div>\r\n<div>[reveal-answer q=\"hjm482\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm482\"](\u20132, 4) is not a solution of the system.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<h2>Classifying a System of Linear Equations in 2 Variables<\/h2>\r\nIn addition to considering the number of equations and variables, we can categorize a system of linear equations by the number of solutions it has. A system of equations is said to be\u00a0<em><strong>consistent <\/strong><\/em>if it\u00a0has at least one solution.\r\n\r\nA consistent system is considered to be\u00a0<strong><em>independent\u00a0<\/em><\/strong>if it has a single solution, like the example we just explored. Graphically, this means that the two lines that make up the system have different slopes and intersect at exactly one point in the plane (e.g., figure 1).\r\n\r\nA consistent system is considered to be <strong><em>dependent\u00a0<\/em><\/strong>if the two lines that make up the system have the same slope and the same [latex]y[\/latex]-intercept. In other words, the lines coincide, so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions of the system that all lie on the same line.\r\n\r\nA system of linear equations is said to be\u00a0<strong>inconsistent <\/strong>if it has no solution. This is where the graph of the system\u00a0consists of two parallel lines. The lines have the same slope and different [latex]y[\/latex]<em>-<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.\r\n<div class=\"textbox\">\r\n<h3>Linear Systems of equations<\/h3>\r\nThere are three types of systems of linear equations in two variables and three types of solutions.\r\n<div>\r\n<ul>\r\n \t<li>A\u00a0<strong>consistent,<\/strong>\u00a0<strong>independent system\u00a0<\/strong>has exactly one solution. Graphically, the point where the two lines intersect is the only solution.<\/li>\r\n \t<li>A <strong>consistent, dependent system\u00a0<\/strong>has infinitely many solutions. Graphically, the lines coincide. Since they are the same line, every coordinate pair on the line is a solution of the system.<\/li>\r\n \t<li>An\u00a0<strong>inconsistent system\u00a0<\/strong>has no solution. Graphically, the two lines are parallel and never intersect.<\/li>\r\n<\/ul>\r\n<\/div>\r\nGraphical representations of each type of system:\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222630\/CNX_Precalc_Figure_09_01_002n2.jpg\" alt=\"Graphs of an independent system, an inconsistent system, and a dependent system as described above. \" width=\"945\" height=\"479\" \/>\r\n\r\nThe independent and dependent systems are also consistent because they both have at least one solution.<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video shows examples of how to identify whether a graphed system has a solution and identify what type of solution is represented.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/ZolxtOjcEQY?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-2.odt\">Transcript 2.6.1 Video 2<\/a>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nClassify the system of equations shown on the graph:\r\n\r\n<img class=\"aligncenter size-medium wp-image-1479\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10235050\/parallel-lines-300x300.png\" alt=\"graph of two parallel lines\" width=\"300\" height=\"300\" \/>\r\n<h4>Solution<\/h4>\r\nThe lines are parallel and never cross, therefore there is no solution.\r\n\r\nThe system is inconsistent.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nClassify the system of equations shown on the graph:\r\n\r\n<img class=\"aligncenter size-medium wp-image-1478\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10234904\/consistent-and-independent-300x300.png\" alt=\"2 lines that cross at one point\" width=\"300\" height=\"300\" \/>\r\n[reveal-answer q=\"hjm237\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm237\"]\r\n\r\nThe system is consistent and independent.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving Systems of Equations by Graphing<\/h2>\r\nThere are multiple methods of solving systems of linear equations. For a\u00a0<em><strong>system of linear equations\u00a0in two variables<\/strong><\/em>, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nSolve the following system of equations by graphing. Identify the type of system.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=-8\\\\ x-y=-1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Solution<\/h4>\r\nTo graph the equations, we can either plot points or convert the equation to slope intercept form, [latex]y=mx+b[\/latex], and use the slope and [latex]y[\/latex]-intercept.\r\n\r\nSolve the first equation for [latex]y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=-8\\\\ y=-2x - 8\\end{array}[\/latex]<\/p>\r\nSo [latex]m=-2[\/latex] and the [latex]y[\/latex]-intercept is (0, \u20138).\r\n\r\nSolve the second equation for [latex]y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x-y=-1\\\\ y=x+1\\end{array}[\/latex]<\/p>\r\nSo [latex]m=1[\/latex] and the [latex]y[\/latex]-intercept is (0, 1).\r\n\r\nGraph both equations on the same set of axes as seen below.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222636\/CNX_Precalc_Figure_09_01_0122.jpg\" alt=\"A graph of two lines. The first is y equals minus 2x minus 8, the second is y equals x+1. They cross at (-3,-2)\" width=\"487\" height=\"316\" \/><\/div>\r\nClearly, the lines intersect at a single point,so the system is independent.\r\n\r\nThe lines appear to intersect at the point [latex]\\left(-3,-2\\right)[\/latex]. We can check to make sure that this is the solution of the system by substituting the ordered pair into each equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}2\\left(-3\\right)+\\left(-2\\right)=-8\\hfill &amp; \\hfill \\\\ \\text{ }-8=-8\\hfill &amp; \\text{True}\\hfill \\\\ \\text{ }\\left(-3\\right)-\\left(-2\\right)=-1\\hfill &amp; \\hfill \\\\ \\text{ }-1=-1\\hfill &amp; \\text{True}\\hfill \\end{array}[\/latex]<\/p>\r\nThe solution of the system is the ordered pair [latex]\\left(-3,-2\\right)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nSolve the system of equations by graphing. Identify the type of system.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=6\\\\ -2x-y=1\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"hjm866\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm866\"]\r\n\r\n<img class=\"aligncenter size-medium wp-image-1470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10204452\/System-parallel-293x300.png\" alt=\"Graph showing parallel lines\" width=\"293\" height=\"300\" \/>\r\n\r\nThe lines are parallel. So the system of equations is inconsistent. There is no solution.\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nGraphing can be used if the system is inconsistent or dependent.\u00a0In both cases, we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.\r\n\r\nThe next video shows how to solve a system of equations by first graphing the lines and then identifying the solution the system has.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/Lv832rXAQ5k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-3.odt\">Transcript 2.6.1 Video 3<\/a>\r\n\r\n<\/div>\r\nDetermining the solution of a system of equations graphically does have its drawbacks. Apart from the time it takes to draw the graph, we could end up with a solution whose coordinates are fractions that we cannot read from the graph. Or the lines could look parallel but actually not be parallel.\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nSolve the system of equations by graphing. Identify the type of system.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+3y=6\\\\ 2x-y=1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Solution<\/h4>\r\nUsing points:\r\n\r\n(0, 2) and (3, 0) lie on the line\u00a0[latex]2x+3y=6[\/latex]\r\n\r\n(0, \u20131) and [latex]\\left (\\frac{1}{2}, 0\\right )[\/latex] lie on the line\u00a0[latex]2x-y=1[\/latex]\r\n\r\nGraph the lines:\r\n\r\n<img class=\"aligncenter wp-image-1471 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-297x300.png\" alt=\"Graph of 2x+3y=6 and 2x-y=1, crossing at a single point, but the coordinates can't be read.\" width=\"297\" height=\"300\" \/>\r\n\r\nThe two lines cross at a single point so there is one solution. This means that the system of equations is consistent and independent.\r\n\r\nHowever, it is impossible to determine the exact solution from the graph as the coordinates are fractional.\r\n\r\nThis is why algebraic methods are needed to solve systems of equations.\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-105\" class=\"standard post-105 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Explain the meaning of a solution or solutions of a system of linear equations<\/li>\n<li>Explain the three types of systems of linear equations<\/li>\n<li>Solve a system of linear equations by graphing<\/li>\n<\/ul>\n<\/div>\n<h2>Systems of Equations<\/h2>\n<p>A\u00a0<strong><em>system of linear equations\u00a0<\/em><\/strong>consists of two or more linear equations in two or more variables such that all equations in the system are considered simultaneously. To find the single, unique solution of a system of linear equations, we must find a numerical value for each variable in the system that satisfies <em>all<\/em> equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a single, unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a single, unique solution.<\/p>\n<p>In this section, we will look at systems of linear equations in two variables, which consist of two equations that each contain the same two variables. For example, consider this system of linear equations in two variables:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}2x+y=\\text{ }15\\\\3x-y=\\text{ }5\\end{array}[\/latex]<\/div>\n<p>The\u00a0<em><strong>solution<\/strong><\/em>\u00a0of a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair [latex](4, 7)[\/latex] is the solution of the system of linear equations. We can verify the solution by substituting the value of each variable into each equation to see if the ordered pair satisfies both equations:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}2\\left(4\\right)+\\left(7\\right)=15\\text{ }\\text{True}\\hfill \\\\ 3\\left(4\\right)-\\left(7\\right)=5\\text{ }\\text{True}\\hfill \\end{array}[\/latex]<\/div>\n<p><span style=\"font-size: 1em;\">This system consists of two linear equations, so graphically, we are considering two lines. Each point on the first line is a solution of the first equation.\u00a0Each point on the second line is a solution of the second equation. Only a point on both lines will satisfy both equations. For the system we just looked at, the two lines cross at the solution (4, 7), as shown in figure 1.<\/span><\/p>\n<\/div>\n<div class=\"entry-content\">\n<div id=\"attachment_1477\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1477\" class=\"wp-image-1477 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-300x300.png\" alt=\"2 lines crossing at a single point, (4,7)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Soln-of-a-system.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1477\" class=\"wp-caption-text\">Figure 1. The intersection point of the two lines is the solution of the system.<\/p>\n<\/div>\n<div id=\"post-105\" class=\"standard post-105 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox\">\n<h3>How To determine whether an ordered pair is a solution of a system<\/h3>\n<ol>\n<li>Substitute the ordered pair into each equation in the system.<\/li>\n<li>Determine whether true statements result from the substitution in each equation; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution of the system of equations:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+3y=8\\hfill \\\\ 2x - 9=y\\hfill \\end{array}[\/latex]<\/div>\n<h4>Solution<\/h4>\n<p>Substitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into each equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\left(5\\right)+3\\left(1\\right)=8\\hfill & \\hfill \\\\ \\text{ }8=8\\hfill & \\text{True}\\hfill \\\\2\\left(5\\right)-9=\\left(1\\right)\\hfill & \\hfill \\\\ \\text{ }\\text{1=1}\\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\n<p>The ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution of the system.<\/p>\n<p>We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222634\/CNX_Precalc_Figure_09_01_0032.jpg\" alt=\"A graph of two lines running through the point five, one. The first line's equation is x plus 3y equals 8. The second line's equation is 2x minus 9 equals y.\" width=\"348\" height=\"261\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of how to verify whether an ordered pair is a solution of a system of equations.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/2IxgKgjX00k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-1.odt\">Transcript 2.6.1 Video 1<\/a><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine whether the ordered pair [latex]\\left(\u20132,4\\right)[\/latex] is a solution of the system of equations:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}-x+2&=y\\\\2x+y&=8\\end{aligned}[\/latex]<\/p>\n<div><\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm482\">Show Answer<\/span><\/p>\n<div id=\"qhjm482\" class=\"hidden-answer\" style=\"display: none\">(\u20132, 4) is not a solution of the system.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Classifying a System of Linear Equations in 2 Variables<\/h2>\n<p>In addition to considering the number of equations and variables, we can categorize a system of linear equations by the number of solutions it has. A system of equations is said to be\u00a0<em><strong>consistent <\/strong><\/em>if it\u00a0has at least one solution.<\/p>\n<p>A consistent system is considered to be\u00a0<strong><em>independent\u00a0<\/em><\/strong>if it has a single solution, like the example we just explored. Graphically, this means that the two lines that make up the system have different slopes and intersect at exactly one point in the plane (e.g., figure 1).<\/p>\n<p>A consistent system is considered to be <strong><em>dependent\u00a0<\/em><\/strong>if the two lines that make up the system have the same slope and the same [latex]y[\/latex]-intercept. In other words, the lines coincide, so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions of the system that all lie on the same line.<\/p>\n<p>A system of linear equations is said to be\u00a0<strong>inconsistent <\/strong>if it has no solution. This is where the graph of the system\u00a0consists of two parallel lines. The lines have the same slope and different [latex]y[\/latex]<em>&#8211;<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.<\/p>\n<div class=\"textbox\">\n<h3>Linear Systems of equations<\/h3>\n<p>There are three types of systems of linear equations in two variables and three types of solutions.<\/p>\n<div>\n<ul>\n<li>A\u00a0<strong>consistent,<\/strong>\u00a0<strong>independent system\u00a0<\/strong>has exactly one solution. Graphically, the point where the two lines intersect is the only solution.<\/li>\n<li>A <strong>consistent, dependent system\u00a0<\/strong>has infinitely many solutions. Graphically, the lines coincide. Since they are the same line, every coordinate pair on the line is a solution of the system.<\/li>\n<li>An\u00a0<strong>inconsistent system\u00a0<\/strong>has no solution. Graphically, the two lines are parallel and never intersect.<\/li>\n<\/ul>\n<\/div>\n<p>Graphical representations of each type of system:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222630\/CNX_Precalc_Figure_09_01_002n2.jpg\" alt=\"Graphs of an independent system, an inconsistent system, and a dependent system as described above.\" width=\"945\" height=\"479\" \/><\/p>\n<p>The independent and dependent systems are also consistent because they both have at least one solution.<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of how to identify whether a graphed system has a solution and identify what type of solution is represented.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/ZolxtOjcEQY?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-2.odt\">Transcript 2.6.1 Video 2<\/a><\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Classify the system of equations shown on the graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1479\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10235050\/parallel-lines-300x300.png\" alt=\"graph of two parallel lines\" width=\"300\" height=\"300\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The lines are parallel and never cross, therefore there is no solution.<\/p>\n<p>The system is inconsistent.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Classify the system of equations shown on the graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1478\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10234904\/consistent-and-independent-300x300.png\" alt=\"2 lines that cross at one point\" width=\"300\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm237\">Show Answer<\/span><\/p>\n<div id=\"qhjm237\" class=\"hidden-answer\" style=\"display: none\">\n<p>The system is consistent and independent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solving Systems of Equations by Graphing<\/h2>\n<p>There are multiple methods of solving systems of linear equations. For a\u00a0<em><strong>system of linear equations\u00a0in two variables<\/strong><\/em>, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Solve the following system of equations by graphing. Identify the type of system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=-8\\\\ x-y=-1\\end{array}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>To graph the equations, we can either plot points or convert the equation to slope intercept form, [latex]y=mx+b[\/latex], and use the slope and [latex]y[\/latex]-intercept.<\/p>\n<p>Solve the first equation for [latex]y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=-8\\\\ y=-2x - 8\\end{array}[\/latex]<\/p>\n<p>So [latex]m=-2[\/latex] and the [latex]y[\/latex]-intercept is (0, \u20138).<\/p>\n<p>Solve the second equation for [latex]y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x-y=-1\\\\ y=x+1\\end{array}[\/latex]<\/p>\n<p>So [latex]m=1[\/latex] and the [latex]y[\/latex]-intercept is (0, 1).<\/p>\n<p>Graph both equations on the same set of axes as seen below.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222636\/CNX_Precalc_Figure_09_01_0122.jpg\" alt=\"A graph of two lines. The first is y equals minus 2x minus 8, the second is y equals x+1. They cross at (-3,-2)\" width=\"487\" height=\"316\" \/><\/div>\n<p>Clearly, the lines intersect at a single point,so the system is independent.<\/p>\n<p>The lines appear to intersect at the point [latex]\\left(-3,-2\\right)[\/latex]. We can check to make sure that this is the solution of the system by substituting the ordered pair into each equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}2\\left(-3\\right)+\\left(-2\\right)=-8\\hfill & \\hfill \\\\ \\text{ }-8=-8\\hfill & \\text{True}\\hfill \\\\ \\text{ }\\left(-3\\right)-\\left(-2\\right)=-1\\hfill & \\hfill \\\\ \\text{ }-1=-1\\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\n<p>The solution of the system is the ordered pair [latex]\\left(-3,-2\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Solve the system of equations by graphing. Identify the type of system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+y=6\\\\ -2x-y=1\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm866\">Show Answer<\/span><\/p>\n<div id=\"qhjm866\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10204452\/System-parallel-293x300.png\" alt=\"Graph showing parallel lines\" width=\"293\" height=\"300\" \/><\/p>\n<p>The lines are parallel. So the system of equations is inconsistent. There is no solution.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Graphing can be used if the system is inconsistent or dependent.\u00a0In both cases, we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.<\/p>\n<p>The next video shows how to solve a system of equations by first graphing the lines and then identifying the solution the system has.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/Lv832rXAQ5k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.6.1-Video-3.odt\">Transcript 2.6.1 Video 3<\/a><\/p>\n<\/div>\n<p>Determining the solution of a system of equations graphically does have its drawbacks. Apart from the time it takes to draw the graph, we could end up with a solution whose coordinates are fractions that we cannot read from the graph. Or the lines could look parallel but actually not be parallel.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Solve the system of equations by graphing. Identify the type of system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x+3y=6\\\\ 2x-y=1\\end{array}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Using points:<\/p>\n<p>(0, 2) and (3, 0) lie on the line\u00a0[latex]2x+3y=6[\/latex]<\/p>\n<p>(0, \u20131) and [latex]\\left (\\frac{1}{2}, 0\\right )[\/latex] lie on the line\u00a0[latex]2x-y=1[\/latex]<\/p>\n<p>Graph the lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1471 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-297x300.png\" alt=\"Graph of 2x+3y=6 and 2x-y=1, crossing at a single point, but the coordinates can't be read.\" width=\"297\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-297x300.png 297w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-768x776.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-1014x1024.png 1014w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-65x66.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-225x227.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution-350x354.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/System-with-fractional-solution.png 1582w\" sizes=\"auto, (max-width: 297px) 100vw, 297px\" \/><\/p>\n<p>The two lines cross at a single point so there is one solution. This means that the system of equations is consistent and independent.<\/p>\n<p>However, it is impossible to determine the exact solution from the graph as the coordinates are fractional.<\/p>\n<p>This is why algebraic methods are needed to solve systems of equations.<\/p>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/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-1084\">\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>Determine if an Ordered Pair is a Solution to a System of Linear Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2IxgKgjX00k\">https:\/\/youtu.be\/2IxgKgjX00k<\/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>Determine the Number of Solutions to a System of Linear Equations From a Graph. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ZolxtOjcEQY\">https:\/\/youtu.be\/ZolxtOjcEQY<\/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>Try It: hjm866; hjm482; hjm237. Figure 1. Example of classification. Example with fractional solution.. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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 2: Solve a System of Equations by Graphing. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Lv832rXAQ5k\">https:\/\/youtu.be\/Lv832rXAQ5k<\/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: 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":422608,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Revised and adapted: Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Determine if an Ordered Pair is a Solution to a System of Linear Equations\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/2IxgKgjX00k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Determine the Number of Solutions to a System of Linear Equations From a Graph\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ZolxtOjcEQY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Solve a System of Equations by Graphing\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Lv832rXAQ5k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Try It: hjm866; hjm482; hjm237. Figure 1. Example of classification. Example with fractional solution.\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1084","chapter","type-chapter","status-publish","hentry"],"part":963,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1084","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1084\/revisions"}],"predecessor-version":[{"id":4680,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1084\/revisions\/4680"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/963"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1084\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1084"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1084"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1084"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1084"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}