{"id":1775,"date":"2023-10-12T00:32:08","date_gmt":"2023-10-12T00:32:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-to-systems-of-nonlinear-equations-and-inequalities\/"},"modified":"2025-11-19T01:03:12","modified_gmt":"2025-11-19T01:03:12","slug":"introduction-to-systems-of-nonlinear-equations-and-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-to-systems-of-nonlinear-equations-and-inequalities\/","title":{"raw":"Nonlinear Systems","rendered":"Nonlinear Systems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Solve a system of nonlinear equations.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nHalley\u2019s Comet\u00a0orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190502\/CNX_Precalc_Figure_09_03_0012.jpg\" alt=\"A comet streaking across a starry sky.\" width=\"488\" height=\"358\" \/> Halley\u2019s Comet (credit: \"NASA Blueshift\"\/Flickr)[\/caption]\r\n\r\nIn this section we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.\r\n<h2>Graph Nonlinear Inequalities and Systems of Nonlinear Inequalities<\/h2>\r\nAll of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the <strong>inequality<\/strong> symbol. As a reminder, here is one to try. Choose the type of line first, then graph it, and click on the side that satisfies the inequality.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19940&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\nNow we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A <strong>nonlinear inequality<\/strong> is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.\r\n\r\nRecall that when the inequality is greater than, [latex]y&gt;a[\/latex], or less than, [latex]y&lt;a,\\text{}[\/latex] the graph is drawn with a dashed line. When the inequality is greater than or equal to, [latex]y\\ge a,\\text{}[\/latex] or less than or equal to, [latex]y\\le a,\\text{}[\/latex] the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190519\/CNX_Precalc_Figure_09_03_0092.jpg\" alt=\"Four parabolas. For y greater than x squared minus 4 the parabola is dotted, and the region above the parabola is shaded. For y greater than or equal to x squared minus 4 the parabola is solid, and the region above it is shaded. For y less than x squared minus 4 the parabola is dotted, and the region below it is shaded. For y less than or equal to x squared minus 4 the parabola is solid, and the region below it is shaded.\" width=\"975\" height=\"469\" \/> (a) an example of [latex]y&gt;a[\/latex]; (b) an example of [latex]y\\ge a[\/latex]; (c) an example of [latex]y&lt;a[\/latex]; (d) an example of [latex]y\\le a[\/latex][\/caption]\r\n<div class=\"textbox\">\r\n<h3>How To: Given an inequality bounded by a parabola, sketch a graph.<\/h3>\r\n<ol>\r\n \t<li>Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.<\/li>\r\n \t<li>If the boundary is included in the region (the operator is [latex]\\le [\/latex] or [latex]\\ge [\/latex] ), the parabola is graphed as a solid line.<\/li>\r\n \t<li>If the boundary is not included in the region (the operator is &lt; or &gt;), the parabola is graphed as a dashed line.<\/li>\r\n \t<li>Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is true, the solution set is the region including the point. If the statement is false, the solution set is the region on the other side of the boundary line.<\/li>\r\n \t<li>Shade the region representing the solution set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing an Inequality for a Parabola<\/h3>\r\nGraph the inequality [latex]y&gt;{x}^{2}+1[\/latex].\r\n\r\n[reveal-answer q=\"403901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"403901\"]\r\n\r\nFirst, graph the corresponding equation [latex]y={x}^{2}+1[\/latex].\r\n\r\nSince [latex]y&gt;{x}^{2}+1[\/latex] has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let\u2019s test the points\u00a0[latex]\\left(0,2\\right)[\/latex] and [latex]\\left(2,0\\right)[\/latex]. One point is clearly inside the parabola and the other point is clearly outside.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;&gt;{x}^{2}+1 \\\\ 2&amp;&gt;{\\left(0\\right)}^{2}+1 \\\\ 2&amp;&gt;1 &amp;&amp;\\text{True} \\\\[5mm] 0&amp;&gt;{\\left(2\\right)}^{2}+1 \\\\ 0&amp;&gt;5 &amp;&amp;\\text{False} \\end{align}[\/latex]<\/p>\r\nThe graph is shown below. We can see that the solution set consists of all points inside the parabola, but not on the graph itself.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190521\/CNX_Precalc_Figure_09_03_0102.jpg\" alt=\"A dotted parabola with the region above it shaded. The point 0,2 is within the shaded region. The point 2,0 is not within the shaded region.\" width=\"487\" height=\"328\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n<iframe id=\"mom11\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=127575&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Graphing a System of Nonlinear Inequalities<\/h2>\r\nNow that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A <strong>system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the <strong>feasible region<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a system of nonlinear inequalities, sketch a graph.<\/h3>\r\n<ol>\r\n \t<li>Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\r\n \t<li>Graph the nonlinear equations.<\/li>\r\n \t<li>Find the shaded regions of each inequality.<\/li>\r\n \t<li>Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a System of Inequalities<\/h3>\r\nGraph the given system of inequalities.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {x}^{2}-y\\le 0\\\\ 2{x}^{2}+y\\le 12\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"801753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"801753\"]\r\n\r\nThese two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable [latex]y[\/latex] will be eliminated. Then we solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align} x^{2}\u2212y&amp;=0 \\\\ 2x^{2}+y&amp;=12 \\\\ \\hline 3x^{2}&amp;=12 \\\\ x^{2}&amp;=4 \\\\ x&amp;=\\pm 2\\end{align}[\/latex]<\/p>\r\nSubstitute the <em>x<\/em>-values into one of the equations and solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} {x}^{2}-y&amp;=0\\\\ {\\left(2\\right)}^{2}-y&amp;=0\\\\ 4-y&amp;=0\\\\ y&amp;=4\\\\[5mm] {\\left(-2\\right)}^{2}-y&amp;=0\\\\ 4-y&amp;=0\\\\ y&amp;=4\\end{align}[\/latex]<\/p>\r\nThe two points of intersection are [latex]\\left(2,4\\right)[\/latex] and [latex]\\left(-2,4\\right)[\/latex]. Notice that the equations can be rewritten as follows.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{x}^{2}-y&amp;\\le 0 \\\\ {x}^{2}&amp;\\le y \\\\ y&amp;\\ge {x}^{2}\\end{align}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 2{x}^{2}+y\\le 12 \\\\ y\\le -2{x}^{2}+12 \\end{gathered}[\/latex]<\/p>\r\nGraph each inequality.\u00a0The feasible region is the region between the two equations bounded by [latex]2{x}^{2}+y\\le 12[\/latex] on the top and [latex]{x}^{2}-y\\le 0[\/latex] on the bottom.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190523\/CNX_Precalc_Figure_09_03_0112.jpg\" alt=\"Two parabolas that intersect at the points negative 2, four and two, four. The region above the orange parabola is shaded, and the region below the blue parabola is shaded.\" width=\"487\" height=\"367\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph the given system of inequalities.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y\\ge {x}^{2}-1 \\\\ x-y\\ge -1 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"689015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689015\"]\r\n\r\nShade the area bounded by the two curves, above the quadratic and below the line.\r\n\r\n<img class=\"aligncenter size-full wp-image-3175\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/19182112\/CNX_Precalc_Figure_09_03_0122.jpg\" alt=\"A line intersecting a parabola at the points negative one, zero and two, three. The region under the line but above the parabola is shaded.\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=70349&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Methods for Solving a System of Nonlinear Equations<\/h2>\r\nA <strong>system of nonlinear equations<\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form [latex]Ax+By+C=0[\/latex]. Any equation that cannot be written in this form in nonlinear. 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.\r\n<h2>Intersection of a Parabola and a Line<\/h2>\r\nThere are three possible types of solutions for a system of nonlinear equations involving a <strong>parabola<\/strong> and a line.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Possible Types of Solutions for Points of Intersection of a Parabola and a Line<\/h3>\r\nThe graphs below\u00a0illustrate possible solution sets for a system of equations involving a parabola and a line.\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190504\/CNX_Precalc_Figure_09_03_002n2.jpg\" alt=\"Graphs described in main body.\" width=\"945\" height=\"389\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a system of equations containing a line and a parabola, find the solution.<strong>\r\n<\/strong><\/h3>\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 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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a System of Nonlinear Equations Representing a Parabola and a Line<\/h3>\r\nSolve the system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x-y=-1 \\\\ y={x}^{2}+1 \\end{gathered}[\/latex]<\/div>\r\n<div style=\"text-align: left;\">[reveal-answer q=\"654580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654580\"]<\/div>\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. \\\\\u00a0 \\\\ &amp;y={x}^{2}+1 \\\\ &amp;y={\\left(y - 1\\right)}^{2}+1 &amp;&amp; \\text{Substitute expression for }x. \\\\[3mm] &amp;y=\\left({y}^{2}-2y+1\\right)+1 &amp;&amp; \\text{Expand} \\\\ &amp;y={y}^{2}-2y+2 \\\\[3mm] 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]. Next, 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>[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\n<div><\/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<img class=\"aligncenter\" 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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>Could we have substituted values for [latex]y[\/latex] into the second equation to solve for [latex]x[\/latex] in Example 1?<\/h3>\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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the given system of equations by substitution.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}3x-y=-2 \\\\ 2{x}^{2}-y=0 \\end{gathered}[\/latex]<\/div>\r\n&nbsp;\r\n<p style=\"text-align: left;\">[reveal-answer q=\"503330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"503330\"]<\/p>\r\n[latex]\\left(-\\frac{1}{2},\\frac{1}{2}\\right)[\/latex] and [latex]\\left(2,8\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65041&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Intersection of a Circle and a Line<\/h2>\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<div class=\"textbox\">\r\n<h3>A General Note: 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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190509\/CNX_Precalc_Figure_09_03_004n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"337\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a system of equations containing a line and a circle, find the solution.<strong>\r\n<\/strong><\/h3>\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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intersection of a Circle and a Line by Substitution<\/h3>\r\nFind 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[reveal-answer q=\"766252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"766252\"]\r\n\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\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<img class=\"aligncenter\" 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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the system of nonlinear equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=10\\hfill \\\\ x - 3y=-10\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"369213\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"369213\"][latex]\\left(-1,3\\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=62704&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Solving a System of Nonlinear Equations Using Elimination<\/h2>\r\nWe 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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse<\/h3>\r\nThe figure below illustrates possible solution sets for a system of equations involving a circle and an <strong>ellipse<\/strong>.\r\n<ul>\r\n \t<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>\r\n \t<li>One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\r\n \t<li>Two solutions. The circle and the ellipse intersect at two points.<\/li>\r\n \t<li>Three solutions. The circle and the ellipse intersect at three points.<\/li>\r\n \t<li>Four solutions. The circle and the ellipse intersect at four points.<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190514\/CNX_Precalc_Figure_09_03_006n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"238\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<\/h3>\r\nSolve the system of nonlinear equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} {x}^{2}+{y}^{2}=26 \\hspace{5mm} \\left(1\\right)\\\\ 3{x}^{2}+25{y}^{2}=100 \\hspace{5mm} \\left(2\\right)\\end{align}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"241390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"241390\"]\r\n\r\nLet\u2019s begin by multiplying equation (1) by -3, and adding it to equation (2).\r\n<div style=\"text-align: center;\">[latex]\\left(-3\\right)\\left({x}^{2}+{y}^{2}\\right)=\\left(-3\\right)\\left(26\\right)[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}-3{x}^{2}-3{y}^{2}&amp;=-78 \\\\ 3{x}^{2}+25{y}^{2}&amp;=100 \\\\ \\hline 22{y}^{2}&amp;=22 \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: left;\">After we add the two equations together, we solve for [latex]y[\/latex].<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{y}^{2}=1 \\\\ &amp;y=\\pm \\sqrt{1}=\\pm 1 \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\nSubstitute [latex]y=\\pm 1[\/latex] into one of the equations and solve for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}^{2}+{\\left(1\\right)}^{2}=26 \\\\ &amp;{x}^{2}+1=26 \\\\ &amp;{x}^{2}=25 \\\\ &amp;x=\\pm \\sqrt{25}=\\pm 5 \\\\ \\\\ &amp;{x}^{2}+{\\left(-1\\right)}^{2}=26 \\\\ &amp;{x}^{2}+1=26 \\\\ &amp;{x}^{2}=25=\\pm 5 \\end{align}[\/latex]<\/div>\r\nThere are four solutions:\r\n<p style=\"text-align: center;\">[latex]\\left(5,1\\right),\\left(-5,1\\right),\\left(5,-1\\right),\\text{and}\\left(-5,-1\\right)[\/latex].<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190516\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the solution set for the given system of nonlinear equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{gathered}[\/latex]<\/div>\r\n[reveal-answer q=\"880387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880387\"][latex]\\left\\{\\left(1,3\\right),\\left(1,-3\\right),\\left(-1,3\\right),\\left(-1,-3\\right)\\right\\}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we present an example of how to solve a system of non-linear equations that represent the intersection of an ellipse and a hyperbola.\r\n\r\nhttps:\/\/youtu.be\/Ic-42kmdJqc\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points.<\/li>\r\n \t<li>There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the parabola; (3) two solutions, the line intersects the circle in two points.<\/li>\r\n \t<li>There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:\r\n(1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points.<\/li>\r\n \t<li>An inequality is graphed in much the same way as an equation, except for &gt; or &lt;, we draw a dashed line and shade the region containing the solution set.<\/li>\r\n \t<li>Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>feasible region<\/strong> the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect\r\n\r\n<strong>nonlinear inequality<\/strong> an inequality containing a nonlinear expression\r\n\r\n<strong>system of nonlinear equations<\/strong> a system of equations containing at least one equation that is of degree larger than one\r\n\r\n<strong>system of nonlinear inequalities<\/strong> a system of two or more inequalities in two or more variables containing at least one inequality that is not linear\r\n\r\n<img class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/>\r\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 2.6 - Nonlinear Systems<\/span><\/h3>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Solve a system of nonlinear equations.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Halley\u2019s Comet\u00a0orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.<\/p>\n<div style=\"width: 498px\" 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\/03190502\/CNX_Precalc_Figure_09_03_0012.jpg\" alt=\"A comet streaking across a starry sky.\" width=\"488\" height=\"358\" \/><\/p>\n<p class=\"wp-caption-text\">Halley\u2019s Comet (credit: &#8220;NASA Blueshift&#8221;\/Flickr)<\/p>\n<\/div>\n<p>In this section we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.<\/p>\n<h2>Graph Nonlinear Inequalities and Systems of Nonlinear Inequalities<\/h2>\n<p>All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the <strong>inequality<\/strong> symbol. As a reminder, here is one to try. Choose the type of line first, then graph it, and click on the side that satisfies the inequality.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19940&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<p>Now we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A <strong>nonlinear inequality<\/strong> is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.<\/p>\n<p>Recall that when the inequality is greater than, [latex]y>a[\/latex], or less than, [latex]y<a,\\text{}[\/latex] the graph is drawn with a dashed line. When the inequality is greater than or equal to, [latex]y\\ge a,\\text{}[\/latex] or less than or equal to, [latex]y\\le a,\\text{}[\/latex] the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade.\n\n\n\n<div style=\"width: 985px\" 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\/03190519\/CNX_Precalc_Figure_09_03_0092.jpg\" alt=\"Four parabolas. For y greater than x squared minus 4 the parabola is dotted, and the region above the parabola is shaded. For y greater than or equal to x squared minus 4 the parabola is solid, and the region above it is shaded. For y less than x squared minus 4 the parabola is dotted, and the region below it is shaded. For y less than or equal to x squared minus 4 the parabola is solid, and the region below it is shaded.\" width=\"975\" height=\"469\" \/><\/p>\n<p class=\"wp-caption-text\">(a) an example of [latex]y&gt;a[\/latex]; (b) an example of [latex]y\\ge a[\/latex]; (c) an example of [latex]y&lt;a[\/latex]; (d) an example of [latex]y\\le a[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an inequality bounded by a parabola, sketch a graph.<\/h3>\n<ol>\n<li>Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.<\/li>\n<li>If the boundary is included in the region (the operator is [latex]\\le[\/latex] or [latex]\\ge[\/latex] ), the parabola is graphed as a solid line.<\/li>\n<li>If the boundary is not included in the region (the operator is &lt; or &gt;), the parabola is graphed as a dashed line.<\/li>\n<li>Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is true, the solution set is the region including the point. If the statement is false, the solution set is the region on the other side of the boundary line.<\/li>\n<li>Shade the region representing the solution set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing an Inequality for a Parabola<\/h3>\n<p>Graph the inequality [latex]y>{x}^{2}+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q403901\">Show Solution<\/span><\/p>\n<div id=\"q403901\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, graph the corresponding equation [latex]y={x}^{2}+1[\/latex].<\/p>\n<p>Since [latex]y>{x}^{2}+1[\/latex] has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let\u2019s test the points\u00a0[latex]\\left(0,2\\right)[\/latex] and [latex]\\left(2,0\\right)[\/latex]. One point is clearly inside the parabola and the other point is clearly outside.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&>{x}^{2}+1 \\\\ 2&>{\\left(0\\right)}^{2}+1 \\\\ 2&>1 &&\\text{True} \\\\[5mm] 0&>{\\left(2\\right)}^{2}+1 \\\\ 0&>5 &&\\text{False} \\end{align}[\/latex]<\/p>\n<p>The graph is shown below. We can see that the solution set consists of all points inside the parabola, but not on the graph itself.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190521\/CNX_Precalc_Figure_09_03_0102.jpg\" alt=\"A dotted parabola with the region above it shaded. The point 0,2 is within the shaded region. The point 2,0 is not within the shaded region.\" width=\"487\" height=\"328\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom11\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=127575&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing a System of Nonlinear Inequalities<\/h2>\n<p>Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A <strong>system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the <strong>feasible region<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a system of nonlinear inequalities, sketch a graph.<\/h3>\n<ol>\n<li>Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\n<li>Graph the nonlinear equations.<\/li>\n<li>Find the shaded regions of each inequality.<\/li>\n<li>Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a System of Inequalities<\/h3>\n<p>Graph the given system of inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {x}^{2}-y\\le 0\\\\ 2{x}^{2}+y\\le 12\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q801753\">Show Solution<\/span><\/p>\n<div id=\"q801753\" class=\"hidden-answer\" style=\"display: none\">\n<p>These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable [latex]y[\/latex] will be eliminated. Then we solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} x^{2}\u2212y&=0 \\\\ 2x^{2}+y&=12 \\\\ \\hline 3x^{2}&=12 \\\\ x^{2}&=4 \\\\ x&=\\pm 2\\end{align}[\/latex]<\/p>\n<p>Substitute the <em>x<\/em>-values into one of the equations and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} {x}^{2}-y&=0\\\\ {\\left(2\\right)}^{2}-y&=0\\\\ 4-y&=0\\\\ y&=4\\\\[5mm] {\\left(-2\\right)}^{2}-y&=0\\\\ 4-y&=0\\\\ y&=4\\end{align}[\/latex]<\/p>\n<p>The two points of intersection are [latex]\\left(2,4\\right)[\/latex] and [latex]\\left(-2,4\\right)[\/latex]. Notice that the equations can be rewritten as follows.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{x}^{2}-y&\\le 0 \\\\ {x}^{2}&\\le y \\\\ y&\\ge {x}^{2}\\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 2{x}^{2}+y\\le 12 \\\\ y\\le -2{x}^{2}+12 \\end{gathered}[\/latex]<\/p>\n<p>Graph each inequality.\u00a0The feasible region is the region between the two equations bounded by [latex]2{x}^{2}+y\\le 12[\/latex] on the top and [latex]{x}^{2}-y\\le 0[\/latex] on the bottom.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190523\/CNX_Precalc_Figure_09_03_0112.jpg\" alt=\"Two parabolas that intersect at the points negative 2, four and two, four. The region above the orange parabola is shaded, and the region below the blue parabola is shaded.\" width=\"487\" height=\"367\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph the given system of inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y\\ge {x}^{2}-1 \\\\ x-y\\ge -1 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689015\">Show Solution<\/span><\/p>\n<div id=\"q689015\" class=\"hidden-answer\" style=\"display: none\">\n<p>Shade the area bounded by the two curves, above the quadratic and below the line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3175\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/19182112\/CNX_Precalc_Figure_09_03_0122.jpg\" alt=\"A line intersecting a parabola at the points negative one, zero and two, three. The region under the line but above the parabola is shaded.\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=70349&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<h2>Methods for Solving a System of Nonlinear Equations<\/h2>\n<p>A <strong>system of nonlinear equations<\/strong> is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form [latex]Ax+By+C=0[\/latex]. Any equation that cannot be written in this form in nonlinear. 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<h2>Intersection of a Parabola and a Line<\/h2>\n<p>There are three possible types of solutions for a system of nonlinear equations involving a <strong>parabola<\/strong> and a line.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Possible Types of Solutions for Points of Intersection of a Parabola and a Line<\/h3>\n<p>The graphs below\u00a0illustrate possible solution sets for a system of equations involving a parabola and a line.<\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190504\/CNX_Precalc_Figure_09_03_002n2.jpg\" alt=\"Graphs described in main body.\" width=\"945\" height=\"389\" \/><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a system of equations containing a line and a parabola, find the solution.<strong><br \/>\n<\/strong><\/h3>\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 parabola equation.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Check your solutions in both equations.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a System of Nonlinear Equations Representing a Parabola and a Line<\/h3>\n<p>Solve the system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x-y=-1 \\\\ y={x}^{2}+1 \\end{gathered}[\/latex]<\/div>\n<div style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654580\">Show Solution<\/span><\/p>\n<div id=\"q654580\" class=\"hidden-answer\" style=\"display: none\"><\/div>\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. \\\\\u00a0 \\\\ &y={x}^{2}+1 \\\\ &y={\\left(y - 1\\right)}^{2}+1 && \\text{Substitute expression for }x. \\\\[3mm] &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]. Next, 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>[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<div><\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Could we have substituted values for [latex]y[\/latex] into the second equation to solve for [latex]x[\/latex] in Example 1?<\/h3>\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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the given system of equations by substitution.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}3x-y=-2 \\\\ 2{x}^{2}-y=0 \\end{gathered}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q503330\">Show Solution<\/span><\/p>\n<div id=\"q503330\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\frac{1}{2},\\frac{1}{2}\\right)[\/latex] and [latex]\\left(2,8\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65041&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Intersection of a Circle and a Line<\/h2>\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<div class=\"textbox\">\n<h3>A General Note: 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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190509\/CNX_Precalc_Figure_09_03_004n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"337\" \/><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a system of equations containing a line and a circle, find the solution.<strong><br \/>\n<\/strong><\/h3>\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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Intersection of a Circle and a Line by Substitution<\/h3>\n<p>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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q766252\">Show Solution<\/span><\/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<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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the system of nonlinear equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=10\\hfill \\\\ x - 3y=-10\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q369213\">Show Solution<\/span><\/p>\n<div id=\"q369213\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(-1,3\\right)[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=62704&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<h2>Solving a System of Nonlinear Equations Using Elimination<\/h2>\n<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.<\/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>The figure below 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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190514\/CNX_Precalc_Figure_09_03_006n2.jpg\" alt=\"Image described in main body\" width=\"945\" height=\"238\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: 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{align} {x}^{2}+{y}^{2}=26 \\hspace{5mm} \\left(1\\right)\\\\ 3{x}^{2}+25{y}^{2}=100 \\hspace{5mm} \\left(2\\right)\\end{align}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q241390\">Show Solution<\/span><\/p>\n<div id=\"q241390\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by multiplying equation (1) by -3, and adding it to equation (2).<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-3\\right)\\left({x}^{2}+{y}^{2}\\right)=\\left(-3\\right)\\left(26\\right)[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}-3{x}^{2}-3{y}^{2}&=-78 \\\\ 3{x}^{2}+25{y}^{2}&=100 \\\\ \\hline 22{y}^{2}&=22 \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: left;\">After we add the two equations together, we solve for [latex]y[\/latex].<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}&{y}^{2}=1 \\\\ &y=\\pm \\sqrt{1}=\\pm 1 \\end{align}[\/latex]<\/div>\n<div><\/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{align}&{x}^{2}+{\\left(1\\right)}^{2}=26 \\\\ &{x}^{2}+1=26 \\\\ &{x}^{2}=25 \\\\ &x=\\pm \\sqrt{25}=\\pm 5 \\\\ \\\\ &{x}^{2}+{\\left(-1\\right)}^{2}=26 \\\\ &{x}^{2}+1=26 \\\\ &{x}^{2}=25=\\pm 5 \\end{align}[\/latex]<\/div>\n<p>There are four solutions:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(5,1\\right),\\left(-5,1\\right),\\left(5,-1\\right),\\text{and}\\left(-5,-1\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03190516\/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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the solution set for the given system of nonlinear equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{gathered}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880387\">Show Solution<\/span><\/p>\n<div id=\"q880387\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left\\{\\left(1,3\\right),\\left(1,-3\\right),\\left(-1,3\\right),\\left(-1,-3\\right)\\right\\}[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present an example of how to solve a system of non-linear equations that represent the intersection of an ellipse and a hyperbola.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 3: Solve a System of Nonlinear Equations (Elimination)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Ic-42kmdJqc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points.<\/li>\n<li>There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the parabola; (3) two solutions, the line intersects the circle in two points.<\/li>\n<li>There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:<br \/>\n(1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points.<\/li>\n<li>An inequality is graphed in much the same way as an equation, except for &gt; or &lt;, we draw a dashed line and shade the region containing the solution set.<\/li>\n<li>Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>feasible region<\/strong> the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect<\/p>\n<p><strong>nonlinear inequality<\/strong> an inequality containing a nonlinear expression<\/p>\n<p><strong>system of nonlinear equations<\/strong> a system of equations containing at least one equation that is of degree larger than one<\/p>\n<p><strong>system of nonlinear inequalities<\/strong> a system of two or more inequalities in two or more variables containing at least one inequality that is not linear<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/><\/p>\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 2.6 &#8211; Nonlinear Systems<\/span><\/h3>\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-1775\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 12575. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 70349. <strong>Authored by<\/strong>: Roy Shahbazian. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 19940. <strong>Authored by<\/strong>: Georgiakaki, Sophia. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question 65041. <strong>Authored by<\/strong>: Etgen, Benjamin. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 62704. <strong>Authored by<\/strong>: Hidegkuti, Marta. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 29703. <strong>Authored by<\/strong>: McClure, Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex 3: Solve a System of Nonlinear Equations (Elimination). <strong>Authored by<\/strong>: Sousa, James (Mathispower4u). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Ic-42kmdJqc\">https:\/\/youtu.be\/Ic-42kmdJqc<\/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>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":708740,"menu_order":11,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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