{"id":1217,"date":"2025-07-23T20:57:21","date_gmt":"2025-07-23T20:57:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1217"},"modified":"2026-03-20T19:07:56","modified_gmt":"2026-03-20T19:07:56","slug":"systems-of-nonlinear-equations-and-inequalities-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-nonlinear-equations-and-inequalities-learn-it-2\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Learn It 2","rendered":"Systems of Nonlinear Equations and Inequalities: Learn It 2"},"content":{"raw":"

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\r\n
\r\n

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 circle<\/strong> and a line.\r\n
    \r\n \t
  • No solution. The line does not intersect the circle.<\/li>\r\n \t
  • One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\r\n \t
  • Two solutions. The line crosses the circle and intersects it at two points.<\/li>\r\n<\/ul>\r\n\"This\r\n\r\n<\/section>
    How To: Given a system of equations containing a line and a circle, find the solution.\r\n<\/strong>\r\n
      \r\n \t
    1. Solve the linear equation for one of the variables.<\/li>\r\n \t
    2. Substitute the expression obtained in step one into the equation for the circle.<\/li>\r\n \t
    3. Solve for the remaining variable.<\/li>\r\n \t
    4. Check your solutions in both equations.<\/li>\r\n<\/ol>\r\n<\/section>
      Find the intersection of the given circle and the given line by substitution.\r\n
      [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\"]<\/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

      [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 \r\n\r\nNow, we factor and solve for [latex]x[\/latex].\r\n
      [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 x<\/em>-values into the original linear equation to solve for [latex]y[\/latex].\r\n
      [latex]\\begin{align}y&=3\\left(2\\right)-5 \\\\ &=1 \\\\[3mm] y&=3\\left(1\\right)-5 \\\\ &=-2 \\end{align}[\/latex]<\/div>\r\n
      \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\"Line\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
      \r\n
      \r\n\r\n[ohm_question hide_question_numbers=1]321627[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section>
      [ohm_question hide_question_numbers=1]321628[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321629[\/ohm_question]<\/section>","rendered":"

      Intersection of a Circle and a Line<\/h2>\n

      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

      \n

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

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

        \n
      • No solution. The line does not intersect the circle.<\/li>\n
      • One solution. The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\n
      • Two solutions. The line crosses the circle and intersects it at two points.<\/li>\n<\/ul>\n

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

        How To: Given a system of equations containing a line and a circle, find the solution.
        \n<\/strong><\/p>\n
          \n
        1. Solve the linear equation for one of the variables.<\/li>\n
        2. Substitute the expression obtained in step one into the equation for the circle.<\/li>\n
        3. Solve for the remaining variable.<\/li>\n
        4. Check your solutions in both equations.<\/li>\n<\/ol>\n<\/section>\n
          Find the intersection of the given circle and the given line by substitution.<\/p>\n
          [latex]\\begin{gathered}{x}^{2}+{y}^{2}=5 \\\\ y=3x - 5 \\end{gathered}[\/latex]<\/div>\n

          \n