{"id":2365,"date":"2024-07-24T23:26:59","date_gmt":"2024-07-24T23:26:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2365"},"modified":"2025-08-15T15:48:36","modified_gmt":"2025-08-15T15:48:36","slug":"systems-of-nonlinear-equations-and-inequalities-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/systems-of-nonlinear-equations-and-inequalities-learn-it-1\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Learn It 1","rendered":"Systems of Nonlinear Equations and Inequalities: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve equations with squared variables or other exponents using substitution and elimination<\/li>\r\n \t<li>Graph curved inequalities and find where they overlap<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\r\nA\u00a0<strong><span id=\"term-00004\" data-type=\"term\">system of nonlinear equations<\/span><\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear.\u00a0Recall that <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">a linear equation can take the form [latex]Ax+By+C = 0[\/latex].<\/span>Any equation that cannot be written in this form in nonlinear.\r\n\r\nThe substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.\r\n<h3>Intersection of a Parabola and a Line<\/h3>\r\n<p id=\"fs-id1165135367535\">There are three possible types of solutions for a system of nonlinear equations involving a\u00a0<span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span> and a line.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>possible types of solutions for points of intersection of a parabola and a line<\/h3>\r\nThe graphs below illustrate possible solution sets for a system of equations involving a parabola (quadratic function) and a line (linear function).\r\n<ul>\r\n \t<li>No solution. The line will never intersect the parabola.<\/li>\r\n \t<li>One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.<\/li>\r\n \t<li>Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_5686\" align=\"aligncenter\" width=\"944\"]<img class=\"wp-image-5686 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5.jpg\" alt=\"The image illustrates three scenarios of intersections between a parabola and a line. In the first graph, there are no intersections, indicating no solutions. The second graph shows the line touching the parabola at a single point, representing one solution. In the third graph, the line intersects the parabola at two points, resulting in two solutions.\" width=\"944\" height=\"325\" \/> Graphs illustrating three scenarios of intersections between a parabola and a line[\/caption]\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p id=\"fs-id1165135694401\"><strong>Given a system of equations containing a line and a parabola, find the solution.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165131884672\" type=\"1\">\r\n \t<li>Solve the linear equation for one of the variables.<\/li>\r\n \t<li>Substitute the expression obtained in step one into the parabola equation.<\/li>\r\n \t<li>Solve for the remaining variable.<\/li>\r\n \t<li>Check your solutions in both equations.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x-y=-1\\hfill \\\\ y={x}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"654580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654580\"]\r\n<div style=\"text-align: left;\">Solve the first equation for [latex]x[\/latex] and then substitute the resulting expression into the second equation.<\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;x-y=-1 \\\\ &amp;x=y - 1 &amp;&amp; \\text{Solve for }x. \\\\ \\\\ &amp;y={x}^{2}+1 \\\\ &amp;y={\\left(y - 1\\right)}^{2}+1 &amp;&amp; \\text{Substitute expression for }x. \\\\ &amp;y=\\left({y}^{2}-2y+1\\right)+1 &amp;&amp; \\text{Expand} \\\\ &amp;y={y}^{2}-2y+2 \\\\[3mm] &amp;0={y}^{2}-3y+2 &amp;&amp; \\text{Set equal to 0 and solve.} \\\\ &amp;0=\\left(y - 2\\right)\\left(y - 1\\right) \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\nSolving for [latex]y[\/latex] gives [latex]y=2[\/latex] and [latex]y=1[\/latex].\r\n[latex]\\\\[\/latex]\r\nNext, substitute each value for [latex]y[\/latex] into the first equation to solve for [latex]x[\/latex]. Always substitute the value into the linear equation to check for extraneous solutions.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x-y=-1 \\\\ x-\\left(2\\right)=-1 \\\\ x=1 \\\\[3mm] x-\\left(1\\right)=-1 \\\\ x=0 \\end{gathered}[\/latex]<\/div>\r\nThe solutions are [latex]\\left(1,2\\right)[\/latex] and [latex]\\left(0,1\\right),\\text{}[\/latex], which can be verified by substituting these [latex]\\left(x,y\\right)[\/latex] values into both of the original equations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190507\/CNX_Precalc_Figure_09_03_0032.jpg\" alt=\"Line x minus y equals negative one crosses parabola y equals x squared plus one at the points zero, one and one, two.\" width=\"487\" height=\"292\" \/> A parabola intersecting with a line at two points[\/caption]\r\n\r\nCould we have substituted values for [latex]y[\/latex] into the second equation to solve for [latex]x[\/latex]?\r\n\r\n<em>Yes, but because [latex]x[\/latex] is squared in the second equation this could give us extraneous solutions for [latex]x[\/latex]. <\/em>\r\n\r\n<em>For<\/em> [latex]y=1[\/latex]\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;y={x}^{2}+1 \\\\ &amp;y={x}^{2}+1 \\\\ &amp;{x}^{2}=0 \\\\ &amp;x=\\pm \\sqrt{0}=0 \\end{align}[\/latex]<\/div>\r\n<em>This gives us the same value as in the solution.<\/em>\r\n\r\n<em>For<\/em> [latex]y=2[\/latex]\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;y={x}^{2}+1 \\\\ &amp;2={x}^{2}+1 \\\\ &amp;{x}^{2}=1 \\\\ &amp;x=\\pm \\sqrt{1}=\\pm 1 \\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<em>Notice that [latex]-1[\/latex] is an extraneous solution.<\/em>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox interact\" aria-label=\"Interact\">Use an online graphing calculator to graph the parabola [latex]y\\ =\\ x^2+2x-3[\/latex]. Now graph the line [latex]ax+by+c\\ =\\ 0[\/latex] and adjust the values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] to find equations for lines that produce\u00a0systems with the following types of solutions:\r\n<ul>\r\n \t<li>One solution<\/li>\r\n \t<li>Two solutions<\/li>\r\n \t<li>No solutions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Intersection of a Circle and a Line<\/h3>\r\nJust as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>possible types of solutions for the points of intersection of a circle and a line<\/h3>\r\nThe graph below illustrates possible solution sets for a system of equations involving a <strong>circle<\/strong> and a line.\r\n<ul>\r\n \t<li>No solution. The line does not intersect the circle.<\/li>\r\n \t<li>One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\r\n \t<li>Two solutions. The line crosses the circle and intersects it at two points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_5690\" align=\"aligncenter\" width=\"717\"]<img class=\"wp-image-5690 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251.jpg\" alt=\"This image shows three cases of intersections between a line and a circle. In the first case, the line does not touch the circle, resulting in no solutions. In the second case, the line is tangent to the circle, touching it at a single point, indicating one solution. In the third case, the line crosses the circle at two points, giving two solutions.\" width=\"717\" height=\"260\" \/> A diagram illustrating three cases of intersections between a line and a circle[\/caption]\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of equations containing a line and a circle, find the solution.\r\n<\/strong>\r\n<ol>\r\n \t<li>Solve the linear equation for one of the variables.<\/li>\r\n \t<li>Substitute the expression obtained in step one into the equation for the circle.<\/li>\r\n \t<li>Solve for the remaining variable.<\/li>\r\n \t<li>Check your solutions in both equations.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the intersection of the given circle and the given line by substitution.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{y}^{2}=5 \\\\ y=3x - 5 \\end{gathered}[\/latex]<\/div>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"766252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"766252\"]<\/p>\r\nOne of the equations has already been solved for [latex]y[\/latex]. We will substitute [latex]y=3x - 5[\/latex] into the equation for the circle.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{\\left(3x - 5\\right)}^{2}=5\\\\ {x}^{2}+9{x}^{2}-30x+25=5\\\\ 10{x}^{2}-30x+20=0\\end{gathered}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nNow, we factor and solve for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}10\\left({x}^{2}-3x+2\\right)=0 \\\\ 10\\left(x - 2\\right)\\left(x - 1\\right)=0 \\\\ x=2 \\hspace{5mm} x=1 \\end{gathered}[\/latex]<\/div>\r\nSubstitute the two <em>x<\/em>-values into the original linear equation to solve for [latex]y[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}y&amp;=3\\left(2\\right)-5 \\\\ &amp;=1 \\\\[3mm] y&amp;=3\\left(1\\right)-5 \\\\ &amp;=-2 \\end{align}[\/latex]<\/div>\r\n<div>\r\n\r\nThe line intersects the circle at [latex]\\left(2,1\\right)[\/latex] and [latex]\\left(1,-2\\right)[\/latex], which can be verified by substituting these [latex]\\left(x,y\\right)[\/latex] values into both of the original equations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190511\/CNX_Precalc_Figure_09_03_0052.jpg\" alt=\"Line y equals 3x minus 5 crosses circle x squared plus y squared equals five at the points 2,1 and 1, negative 2.\" width=\"487\" height=\"367\" \/> A circle intersecting with a line at two points[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]291803[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve equations with squared variables or other exponents using substitution and elimination<\/li>\n<li>Graph curved inequalities and find where they overlap<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\n<p>A\u00a0<strong><span id=\"term-00004\" data-type=\"term\">system of nonlinear equations<\/span><\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear.\u00a0Recall that <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">a linear equation can take the form [latex]Ax+By+C = 0[\/latex].<\/span>Any equation that cannot be written in this form in nonlinear.<\/p>\n<p>The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.<\/p>\n<h3>Intersection of a Parabola and a Line<\/h3>\n<p id=\"fs-id1165135367535\">There are three possible types of solutions for a system of nonlinear equations involving a\u00a0<span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span> and a line.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>possible types of solutions for points of intersection of a parabola and a line<\/h3>\n<p>The graphs below illustrate possible solution sets for a system of equations involving a parabola (quadratic function) and a line (linear function).<\/p>\n<ul>\n<li>No solution. The line will never intersect the parabola.<\/li>\n<li>One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.<\/li>\n<li>Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.<\/li>\n<\/ul>\n<figure id=\"attachment_5686\" aria-describedby=\"caption-attachment-5686\" style=\"width: 944px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5686 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5.jpg\" alt=\"The image illustrates three scenarios of intersections between a parabola and a line. In the first graph, there are no intersections, indicating no solutions. The second graph shows the line touching the parabola at a single point, representing one solution. In the third graph, the line intersects the parabola at two points, resulting in two solutions.\" width=\"944\" height=\"325\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5.jpg 944w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5-300x103.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5-768x264.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5-65x22.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5-225x77.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28162507\/178eac8e3b6e24d899b9de23cd5d79940d7592c5-350x120.jpg 350w\" sizes=\"(max-width: 944px) 100vw, 944px\" \/><figcaption id=\"caption-attachment-5686\" class=\"wp-caption-text\">Graphs illustrating three scenarios of intersections between a parabola and a line<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p id=\"fs-id1165135694401\"><strong>Given a system of equations containing a line and a parabola, find the solution.<\/strong><\/p>\n<ol id=\"fs-id1165131884672\" type=\"1\">\n<li>Solve the linear equation for one of the variables.<\/li>\n<li>Substitute the expression obtained in step one into the parabola equation.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Check your solutions in both equations.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x-y=-1\\hfill \\\\ y={x}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q654580\">Show Solution<\/button><\/p>\n<div id=\"q654580\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: left;\">Solve the first equation for [latex]x[\/latex] and then substitute the resulting expression into the second equation.<\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}&x-y=-1 \\\\ &x=y - 1 && \\text{Solve for }x. \\\\ \\\\ &y={x}^{2}+1 \\\\ &y={\\left(y - 1\\right)}^{2}+1 && \\text{Substitute expression for }x. \\\\ &y=\\left({y}^{2}-2y+1\\right)+1 && \\text{Expand} \\\\ &y={y}^{2}-2y+2 \\\\[3mm] &0={y}^{2}-3y+2 && \\text{Set equal to 0 and solve.} \\\\ &0=\\left(y - 2\\right)\\left(y - 1\\right) \\end{align}[\/latex]<\/div>\n<div><\/div>\n<p>Solving for [latex]y[\/latex] gives [latex]y=2[\/latex] and [latex]y=1[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nNext, substitute each value for [latex]y[\/latex] into the first equation to solve for [latex]x[\/latex]. Always substitute the value into the linear equation to check for extraneous solutions.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x-y=-1 \\\\ x-\\left(2\\right)=-1 \\\\ x=1 \\\\[3mm] x-\\left(1\\right)=-1 \\\\ x=0 \\end{gathered}[\/latex]<\/div>\n<p>The solutions are [latex]\\left(1,2\\right)[\/latex] and [latex]\\left(0,1\\right),\\text{}[\/latex], which can be verified by substituting these [latex]\\left(x,y\\right)[\/latex] values into both of the original equations.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190507\/CNX_Precalc_Figure_09_03_0032.jpg\" alt=\"Line x minus y equals negative one crosses parabola y equals x squared plus one at the points zero, one and one, two.\" width=\"487\" height=\"292\" \/><figcaption class=\"wp-caption-text\">A parabola intersecting with a line at two points<\/figcaption><\/figure>\n<p>Could we have substituted values for [latex]y[\/latex] into the second equation to solve for [latex]x[\/latex]?<\/p>\n<p><em>Yes, but because [latex]x[\/latex] is squared in the second equation this could give us extraneous solutions for [latex]x[\/latex]. <\/em><\/p>\n<p><em>For<\/em> [latex]y=1[\/latex]<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align}&y={x}^{2}+1 \\\\ &y={x}^{2}+1 \\\\ &{x}^{2}=0 \\\\ &x=\\pm \\sqrt{0}=0 \\end{align}[\/latex]<\/div>\n<p><em>This gives us the same value as in the solution.<\/em><\/p>\n<p><em>For<\/em> [latex]y=2[\/latex]<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&y={x}^{2}+1 \\\\ &2={x}^{2}+1 \\\\ &{x}^{2}=1 \\\\ &x=\\pm \\sqrt{1}=\\pm 1 \\end{align}[\/latex]<\/div>\n<\/div>\n<p><em>Notice that [latex]-1[\/latex] is an extraneous solution.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">Use an online graphing calculator to graph the parabola [latex]y\\ =\\ x^2+2x-3[\/latex]. Now graph the line [latex]ax+by+c\\ =\\ 0[\/latex] and adjust the values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] to find equations for lines that produce\u00a0systems with the following types of solutions:<\/p>\n<ul>\n<li>One solution<\/li>\n<li>Two solutions<\/li>\n<li>No solutions<\/li>\n<\/ul>\n<\/section>\n<h3>Intersection of a Circle and a Line<\/h3>\n<p>Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>possible types of solutions for the points of intersection of a circle and a line<\/h3>\n<p>The graph below illustrates possible solution sets for a system of equations involving a <strong>circle<\/strong> and a line.<\/p>\n<ul>\n<li>No solution. The line does not intersect the circle.<\/li>\n<li>One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\n<li>Two solutions. The line crosses the circle and intersects it at two points.<\/li>\n<\/ul>\n<figure id=\"attachment_5690\" aria-describedby=\"caption-attachment-5690\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5690 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251.jpg\" alt=\"This image shows three cases of intersections between a line and a circle. In the first case, the line does not touch the circle, resulting in no solutions. In the second case, the line is tangent to the circle, touching it at a single point, indicating one solution. In the third case, the line crosses the circle at two points, giving two solutions.\" width=\"717\" height=\"260\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251.jpg 717w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251-300x109.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251-65x24.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251-225x82.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/28163244\/dc6316c2933c3aad2b12f0bf07560cf82f2d7251-350x127.jpg 350w\" sizes=\"(max-width: 717px) 100vw, 717px\" \/><figcaption id=\"caption-attachment-5690\" class=\"wp-caption-text\">A diagram illustrating three cases of intersections between a line and a circle<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of equations containing a line and a circle, find the solution.<br \/>\n<\/strong><\/p>\n<ol>\n<li>Solve the linear equation for one of the variables.<\/li>\n<li>Substitute the expression obtained in step one into the equation for the circle.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Check your solutions in both equations.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the intersection of the given circle and the given line by substitution.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{y}^{2}=5 \\\\ y=3x - 5 \\end{gathered}[\/latex]<\/div>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q766252\">Show Solution<\/button><\/p>\n<div id=\"q766252\" class=\"hidden-answer\" style=\"display: none\">\n<p>One of the equations has already been solved for [latex]y[\/latex]. We will substitute [latex]y=3x - 5[\/latex] into the equation for the circle.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{\\left(3x - 5\\right)}^{2}=5\\\\ {x}^{2}+9{x}^{2}-30x+25=5\\\\ 10{x}^{2}-30x+20=0\\end{gathered}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Now, we factor and solve for [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}10\\left({x}^{2}-3x+2\\right)=0 \\\\ 10\\left(x - 2\\right)\\left(x - 1\\right)=0 \\\\ x=2 \\hspace{5mm} x=1 \\end{gathered}[\/latex]<\/div>\n<p>Substitute the two <em>x<\/em>-values into the original linear equation to solve for [latex]y[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}y&=3\\left(2\\right)-5 \\\\ &=1 \\\\[3mm] y&=3\\left(1\\right)-5 \\\\ &=-2 \\end{align}[\/latex]<\/div>\n<div>\n<p>The line intersects the circle at [latex]\\left(2,1\\right)[\/latex] and [latex]\\left(1,-2\\right)[\/latex], which can be verified by substituting these [latex]\\left(x,y\\right)[\/latex] values into both of the original equations.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190511\/CNX_Precalc_Figure_09_03_0052.jpg\" alt=\"Line y equals 3x minus 5 crosses circle x squared plus y squared equals five at the points 2,1 and 1, negative 2.\" width=\"487\" height=\"367\" \/><figcaption class=\"wp-caption-text\">A circle intersecting with a line at two points<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm291803\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=291803&theme=lumen&iframe_resize_id=ohm291803&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":300,"module-header":"learn_it","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2365"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2365\/revisions"}],"predecessor-version":[{"id":7858,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2365\/revisions\/7858"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/300"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2365\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2365"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2365"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2365"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}