{"id":1761,"date":"2015-11-12T18:30:45","date_gmt":"2015-11-12T18:30:45","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1761"},"modified":"2015-11-12T18:30:45","modified_gmt":"2015-11-12T18:30:45","slug":"solving-a-system-of-nonlinear-equations-using-elimination","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solving-a-system-of-nonlinear-equations-using-elimination\/","title":{"raw":"Solving a System of Nonlinear Equations Using Elimination","rendered":"Solving a System of Nonlinear Equations Using Elimination"},"content":{"raw":"<p>We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, <strong>elimination<\/strong> is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a <strong>circle<\/strong> and an ellipse.\n<\/p><div class=\"textbox\">\n<h3>A General Note: Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse<\/h3>\nFigure 6 illustrates possible solution sets for a system of equations involving a circle and an <strong>ellipse<\/strong>.\n<ul><li>No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.<\/li>\n\t<li>One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\n\t<li>Two solutions. The circle and the ellipse intersect at two points.<\/li>\n\t<li>Three solutions. The circle and the ellipse intersect at three points.<\/li>\n\t<li>Four solutions. The circle and the ellipse intersect at four points.<\/li>\n<\/ul>\n[caption id=\"\" align=\"aligncenter\" width=\"945\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202125\/CNX_Precalc_Figure_09_03_006n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"238\" data-media-type=\"image\/jpg\"\/><b>Figure 6<\/b>[\/caption]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<\/h3>\nSolve the system of nonlinear equations.\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}\\hfill {x}^{2}+{y}^{2}=26&amp; \\hfill \\left(1\\right)\\\\ \\hfill 3{x}^{2}+25{y}^{2}=100&amp; \\hfill \\left(2\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\nLet\u2019s begin by multiplying equation (1) by [latex]-3,\\text{}[\/latex] and adding it to equation (2).\n<div style=\"text-align: center;\">[latex]\\frac{\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\left(-3\\right)\\left({x}^{2}+{y}^{2}\\right)=\\left(-3\\right)\\left(26\\right)\\hfill \\\\ \\text{ }-3{x}^{2}-3{y}^{2}=-78\\hfill \\end{array}\\hfill \\\\ \\text{ }3{x}^{2}+25{y}^{2}=100\\hfill \\end{array}}{\\text{ }22{y}^{2}=22}[\/latex]<\/div>\nAfter we add the two equations together, we solve for [latex]y[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{y}^{2}=1\\hfill \\\\ y=\\pm \\sqrt{1}=\\pm 1\\hfill \\end{array}[\/latex]<\/div>\nSubstitute [latex]y=\\pm 1[\/latex] into one of the equations and solve for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{x}^{2}+{\\left(1\\right)}^{2}=26\\hfill \\\\ \\text{ }{x}^{2}+1=26\\hfill \\\\ \\text{ }{x}^{2}=25\\hfill \\\\ \\text{ }x=\\pm \\sqrt{25}=\\pm 5\\hfill \\\\ \\hfill \\\\ {x}^{2}+{\\left(-1\\right)}^{2}=26\\hfill \\\\ \\text{ }{x}^{2}+1=26\\hfill \\\\ \\text{ }{x}^{2}=25=\\pm 5\\hfill \\end{array}[\/latex]<\/div>\nThere are four solutions: [latex]\\left(5,1\\right),\\left(-5,1\\right),\\left(5,-1\\right),\\text{and}\\left(-5,-1\\right)[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202127\/CNX_Precalc_Figure_09_03_0072.jpg\" alt=\"Circle intersected by ellipse at four points. Those points are negative five, one; five, one; five, negative one; and negative five, negative one.\" width=\"731\" height=\"517\" data-media-type=\"image\/jpg\"\/><b>Figure 7<\/b>[\/caption]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/h3>\nFind the solution set for the given system of nonlinear equations.\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{array}[\/latex]<\/div>\n<a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-19\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>","rendered":"<p>We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, <strong>elimination<\/strong> is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a <strong>circle<\/strong> and an ellipse.\n<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse<\/h3>\n<p>Figure 6 illustrates possible solution sets for a system of equations involving a circle and an <strong>ellipse<\/strong>.<\/p>\n<ul>\n<li>No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.<\/li>\n<li>One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\n<li>Two solutions. The circle and the ellipse intersect at two points.<\/li>\n<li>Three solutions. The circle and the ellipse intersect at three points.<\/li>\n<li>Four solutions. The circle and the ellipse intersect at four points.<\/li>\n<\/ul>\n<div style=\"width: 955px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202125\/CNX_Precalc_Figure_09_03_006n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"238\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<\/h3>\n<p>Solve the system of nonlinear equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}\\hfill {x}^{2}+{y}^{2}=26& \\hfill \\left(1\\right)\\\\ \\hfill 3{x}^{2}+25{y}^{2}=100& \\hfill \\left(2\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\n<p>Let\u2019s begin by multiplying equation (1) by [latex]-3,\\text{}[\/latex] and adding it to equation (2).<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\left(-3\\right)\\left({x}^{2}+{y}^{2}\\right)=\\left(-3\\right)\\left(26\\right)\\hfill \\\\ \\text{ }-3{x}^{2}-3{y}^{2}=-78\\hfill \\end{array}\\hfill \\\\ \\text{ }3{x}^{2}+25{y}^{2}=100\\hfill \\end{array}}{\\text{ }22{y}^{2}=22}[\/latex]<\/div>\n<p>After we add the two equations together, we solve for [latex]y[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{y}^{2}=1\\hfill \\\\ y=\\pm \\sqrt{1}=\\pm 1\\hfill \\end{array}[\/latex]<\/div>\n<p>Substitute [latex]y=\\pm 1[\/latex] into one of the equations and solve for [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{x}^{2}+{\\left(1\\right)}^{2}=26\\hfill \\\\ \\text{ }{x}^{2}+1=26\\hfill \\\\ \\text{ }{x}^{2}=25\\hfill \\\\ \\text{ }x=\\pm \\sqrt{25}=\\pm 5\\hfill \\\\ \\hfill \\\\ {x}^{2}+{\\left(-1\\right)}^{2}=26\\hfill \\\\ \\text{ }{x}^{2}+1=26\\hfill \\\\ \\text{ }{x}^{2}=25=\\pm 5\\hfill \\end{array}[\/latex]<\/div>\n<p>There are four solutions: [latex]\\left(5,1\\right),\\left(-5,1\\right),\\left(5,-1\\right),\\text{and}\\left(-5,-1\\right)[\/latex].<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202127\/CNX_Precalc_Figure_09_03_0072.jpg\" alt=\"Circle intersected by ellipse at four points. Those points are negative five, one; five, one; five, negative one; and negative five, negative one.\" width=\"731\" height=\"517\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/h3>\n<p>Find the solution set for the given system of nonlinear equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{array}[\/latex]<\/div>\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-19\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\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-1761\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":3,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1761","chapter","type-chapter","status-publish","hentry"],"part":1751,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1761","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1761\/revisions"}],"predecessor-version":[{"id":2260,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1761\/revisions\/2260"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1751"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1761\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1761"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1761"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1761"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1761"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}