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Solve the system of equations.\newliney=−27x+24y = -27x + 24\newliney=12x2−27x−24y = 12x^2 - 27x - 24\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=−27x+24y = -27x + 24\newliney=12x2−27x−24y = 12x^2 - 27x - 24\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: Set the two equations equal to each other to find the xx-values where they intersect.y=−27x+24y = -27x + 24y=12x2−27x−24y = 12x^2 - 27x - 24−27x+24=12x2−27x−24-27x + 24 = 12x^2 - 27x - 24
  2. Simplify Equation: Simplify the equation by moving all terms to one side to form a standard quadratic equation.\newline−27x+24−(−27x+24)=12x2−27x−24−(−27x+24)-27x + 24 - (-27x + 24) = 12x^2 - 27x - 24 - (-27x + 24)\newline0=12x2−480 = 12x^2 - 48
  3. Divide and Simplify: Divide the equation by 1212 to simplify it further.\newline0=12x2−480 = 12x^2 - 48\newline0=x2−40 = x^2 - 4
  4. Factor Quadratic Equation: Factor the quadratic equation. x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2)
  5. Solve for x: Solve for x by setting each factor equal to zero.\newline(x−2)=0(x - 2) = 0 or (x+2)=0(x + 2) = 0\newlinex=2x = 2 or x=−2x = -2
  6. Substitute and Find yy: Substitute the xx-values back into one of the original equations to find the corresponding yy-values.\newlineFor x=2x = 2:\newliney=−27(2)+24y = -27(2) + 24\newliney=−54+24y = -54 + 24\newliney=−30y = -30\newlineFor x=−2x = -2:\newliney=−27(−2)+24y = -27(-2) + 24\newliney=54+24y = 54 + 24\newlinexx00
  7. Write Coordinates: Write the coordinates in exact form.\newlineFirst Coordinate: (2,−30)(2, -30)\newlineSecond Coordinate: (−2,78)(-2, 78)

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