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Solve the system of equations.\newliney=−14x+75y = -14x + 75\newliney=x2−14x−46y = x^2 - 14x - 46\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Solve a system of linear and quadratic equations: parabolasright chevron icon

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Q. Solve the system of equations.\newliney=−14x+75y = -14x + 75\newliney=x2−14x−46y = x^2 - 14x - 46\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=−14x+75y = -14x + 75\newliney=x2−14x−46y = x^2 - 14x - 46\newlineSet the two equations equal to each other to find the xx-values where their yy-values are the same.\newline−14x+75=x2−14x−46-14x + 75 = x^2 - 14x - 46
  2. Simplify Quadratic Equation: Simplify the equation by moving all terms to one side to form a quadratic equation.\newline0=x2−14x−46+14x−750 = x^2 - 14x - 46 + 14x - 75\newline0=x2−1210 = x^2 - 121
  3. Solve Quadratic Equation: Solve the quadratic equation. \newlinex2=121x^2 = 121\newlineTake the square root of both sides.\newlinex=±11x = \pm11
  4. Substitute for Y-Values: Find the corresponding yy-values for each xx-value by substituting back into one of the original equations. Let's use y=−14x+75y = -14x + 75. For x=11x = 11: y=−14(11)+75y = -14(11) + 75 y=−154+75y = -154 + 75 y=−79y = -79
  5. Find Y-Value: Find the y-value for the second x-value.\newlineFor x=−11x = -11:\newliney=−14(−11)+75y = -14(-11) + 75\newliney=154+75y = 154 + 75\newliney=229y = 229
  6. Write Coordinates: Write the coordinates in exact form.\newlineThe solutions to the system of equations are the points where the two graphs intersect.\newlineFirst Coordinate: (11,−79)(11, -79)\newlineSecond Coordinate: (−11,229)(-11, 229)

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