{"id":1216,"date":"2025-07-23T20:57:24","date_gmt":"2025-07-23T20:57:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1216"},"modified":"2026-03-20T19:12:11","modified_gmt":"2026-03-20T19:12:11","slug":"systems-of-nonlinear-equations-and-inequalities-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-nonlinear-equations-and-inequalities-learn-it-3\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Learn It 3","rendered":"Systems of Nonlinear Equations and Inequalities: Learn It 3"},"content":{"raw":"

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, 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.\r\n\r\n
It is not necessary that you know the form of an equation given in a system in order to solve the system (such as with an ellipse or hyperbola in the examples below) . If you are unable to sketch the graph of an equation given in the system, you must be extra diligent with your algebra though to avoid missing or extraneous solutions.<\/section>
\r\n

ellipse<\/h3>\r\nAn ellipse is defined as the set of all points where the sum of the distances from two fixed points (the foci) is a constant. It is an \"elongated\" or \"squashed\" circle, characterized by a center, a major axis (the longest diameter) and a minor axis (the shortest diameter) perpendicular to the major axis. A circle is a special case of an ellipse where the two foci are at the same location\r\n\r\n<\/section><\/section>
\r\n

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 ellipse<\/strong>.\r\n
    \r\n \t
  • 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
  • One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\r\n \t
  • Two solutions. The circle and the ellipse intersect at two points.<\/li>\r\n \t
  • Three solutions. The circle and the ellipse intersect at three points.<\/li>\r\n \t
  • Four solutions. The circle and the ellipse intersect at four points.<\/li>\r\n<\/ul>\r\n\"This\r\n\r\n<\/section>
    Solve the system of nonlinear equations.\r\n
    [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 \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 [latex]-3[\/latex], and adding it to equation (2).\r\n
    \r\n
    [latex]\\left(-3\\right)\\left({x}^{2}+{y}^{2}\\right)=\\left(-3\\right)\\left(26\\right)[\/latex]<\/div>\r\n
    <\/div>\r\n
    [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>\r\n<\/div>\r\n
    <\/div>\r\nAfter we add the two equations together, we solve for [latex]y[\/latex].\r\n
    <\/div>\r\n
    [latex]\\begin{align}&{y}^{2}=1 \\\\ &y=\\pm \\sqrt{1}=\\pm 1 \\end{align}[\/latex]<\/div>\r\n
    <\/div>\r\nSubstitute [latex]y=\\pm 1[\/latex] into one of the equations and solve for [latex]x[\/latex].\r\n
    [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>\r\nThere are four solutions:\r\n

    [latex]\\left(5,1\\right),\\left(-5,1\\right),\\left(5,-1\\right),\\text{and}\\left(-5,-1\\right)[\/latex].<\/p>\r\n\"Circle\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321630[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321631[\/ohm_question]<\/section>","rendered":"

    Solving a System of Nonlinear Equations Using Elimination<\/h2>\n

    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, 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.<\/p>\n

    It is not necessary that you know the form of an equation given in a system in order to solve the system (such as with an ellipse or hyperbola in the examples below) . If you are unable to sketch the graph of an equation given in the system, you must be extra diligent with your algebra though to avoid missing or extraneous solutions.<\/section>\n
    \n
    \n

    ellipse<\/h3>\n

    An ellipse is defined as the set of all points where the sum of the distances from two fixed points (the foci) is a constant. It is an “elongated” or “squashed” circle, characterized by a center, a major axis (the longest diameter) and a minor axis (the shortest diameter) perpendicular to the major axis. A circle is a special case of an ellipse where the two foci are at the same location<\/p>\n<\/section>\n<\/section>\n

    \n

    possible types of solutions for the points of intersection of a circle and an ellipse<\/h3>\n

    The figure below illustrates possible solution sets for a system of equations involving a circle and an ellipse<\/strong>.<\/p>\n

      \n
    • 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
    • One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.<\/li>\n
    • Two solutions. The circle and the ellipse intersect at two points.<\/li>\n
    • Three solutions. The circle and the ellipse intersect at three points.<\/li>\n
    • Four solutions. The circle and the ellipse intersect at four points.<\/li>\n<\/ul>\n

      \"This<\/p>\n<\/section>\n

      Solve the system of nonlinear equations.<\/p>\n
      [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>\n