{"id":132,"date":"2025-02-13T22:44:08","date_gmt":"2025-02-13T22:44:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-two-variables\/"},"modified":"2026-03-20T15:13:19","modified_gmt":"2026-03-20T15:13:19","slug":"systems-of-linear-equations-two-variables","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-two-variables\/","title":{"raw":"Systems of Linear Equations in Two Variables: Learn It 1","rendered":"Systems of Linear Equations in Two Variables: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve systems of equations by graphing.<\/li>\r\n \t<li>Solve systems of equations algebraically.<\/li>\r\n \t<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\r\n \t<li>Solve applied systems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solutions of Systems Overview<\/h2>\r\nA <strong>system of linear equations<\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all 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 unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>system of linear equations<\/h3>\r\nA <strong>system of linear equations<\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.\r\n\r\n<\/section>We will start by looking at systems of linear equations in two variables, which consist of two equations that contain two different variables.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the following system of linear equations in two variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}2x+y&amp;=15\\\\[1mm] 3x-y&amp;=5\\end{align}[\/latex]<\/p>\r\nThe <em>solution<\/em> to 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 to the system of linear equations.\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nWe can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}2\\left(4\\right)+\\left(7\\right)&amp;=15 &amp;&amp;\\text{True} \\\\[1mm] 3\\left(4\\right)-\\left(7\\right)&amp;=5 &amp;&amp;\\text{True} \\end{align}[\/latex]<\/p>\r\n\r\n<\/section>In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>types of linear systems<\/h3>\r\nThere are three types of systems of linear equations in two variables, and three types of solutions.\r\n<ul>\r\n \t<li>An <strong>independent<\/strong> system has exactly one solution pair [latex](x,y)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\r\n \t<li>An <strong>inconsistent<\/strong> system has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\r\n \t<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A <\/span><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">dependent<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> system has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/span><\/li>\r\n<\/ul>\r\n<img class=\"aligncenter size-full wp-image-2316\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/22210315\/Screenshot-2024-07-22-at-2.03.07%E2%80%AFPM.png\" alt=\"\" width=\"1316\" height=\"508\" \/>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.<\/strong>\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 both equations; if so, the ordered pair is a solution.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution to the given system of equations.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x+3y&amp;=8\\\\ 2x-9&amp;=y \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"899056\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"899056\"]\r\n\r\nSubstitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into both equations.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(5\\right)+3\\left(1\\right)&amp;=8 \\\\[1mm] 8&amp;=8 &amp;&amp;\\text{True} \\\\[3mm] 2\\left(5\\right)-9&amp;=\\left(1\\right) \\\\[1mm] 1&amp;=1 &amp;&amp;\\text{True} \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution to the system.<\/p>\r\n<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03183603\/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=\"300\" height=\"225\" \/>\r\n\r\n<strong>Analysis of the Solution<\/strong>\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[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321585[\/ohm_question]<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve systems of equations by graphing.<\/li>\n<li>Solve systems of equations algebraically.<\/li>\n<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\n<li>Solve applied systems.<\/li>\n<\/ul>\n<\/section>\n<h2>Solutions of Systems Overview<\/h2>\n<p>A <strong>system of linear equations<\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all 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 unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>system of linear equations<\/h3>\n<p>A <strong>system of linear equations<\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.<\/p>\n<\/section>\n<p>We will start by looking at systems of linear equations in two variables, which consist of two equations that contain two different variables.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the following system of linear equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}2x+y&=15\\\\[1mm] 3x-y&=5\\end{align}[\/latex]<\/p>\n<p>The <em>solution<\/em> to 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 to the system of linear equations.<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}2\\left(4\\right)+\\left(7\\right)&=15 &&\\text{True} \\\\[1mm] 3\\left(4\\right)-\\left(7\\right)&=5 &&\\text{True} \\end{align}[\/latex]<\/p>\n<\/section>\n<p>In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>types of linear systems<\/h3>\n<p>There are three types of systems of linear equations in two variables, and three types of solutions.<\/p>\n<ul>\n<li>An <strong>independent<\/strong> system has exactly one solution pair [latex](x,y)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\n<li>An <strong>inconsistent<\/strong> system has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\n<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A <\/span><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">dependent<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> system has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/span><\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2316\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/22210315\/Screenshot-2024-07-22-at-2.03.07%E2%80%AFPM.png\" alt=\"\" width=\"1316\" height=\"508\" \/><\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.<\/strong><\/p>\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 both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution to the given system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x+3y&=8\\\\ 2x-9&=y \\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q899056\">Show Solution<\/button><\/p>\n<div id=\"q899056\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into both equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(5\\right)+3\\left(1\\right)&=8 \\\\[1mm] 8&=8 &&\\text{True} \\\\[3mm] 2\\left(5\\right)-9&=\\left(1\\right) \\\\[1mm] 1&=1 &&\\text{True} \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left;\">The ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution to the system.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03183603\/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=\"300\" height=\"225\" \/><\/p>\n<p><strong>Analysis of the Solution<\/strong><\/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<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321585\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321585&theme=lumen&iframe_resize_id=ohm321585&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"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":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/132"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions"}],"predecessor-version":[{"id":5928,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions\/5928"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/132\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=132"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=132"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=132"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}