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Solve the system of equations.\newliney=x2−39x−43y = x^2 - 39x - 43\newliney=−39x+101y = -39x + 101\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=x2−39x−43y = x^2 - 39x - 43\newliney=−39x+101y = -39x + 101\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=x2−39x−43y = x^2 - 39x - 43\newliney=−39x+101y = -39x + 101\newlineSet the two equations equal to each other to find the xx-values where they intersect.\newlinex2−39x−43=−39x+101x^2 - 39x - 43 = -39x + 101
  2. Simplify and Standardize: Simplify the equation by adding 39x39x to both sides and subtracting 101101 from both sides to get the quadratic equation in standard form.\newlinex2−39x−43+39x−101=−39x+101+39x−101x^2 - 39x - 43 + 39x - 101 = -39x + 101 + 39x - 101\newlinex2−144=0x^2 - 144 = 0
  3. Solve Quadratic Equation: Solve the quadratic equation for xx.x2−144=0x^2 - 144 = 0Add 144144 to both sides.x2=144x^2 = 144Take the square root of both sides.x=±12x = \pm12
  4. Find y-values: Find the corresponding yy-values for each xx-value by substituting xx back into one of the original equations. We can use the second equation y=−39x+101y = -39x + 101 for simplicity.\newlineFor x=12x = 12:\newliney=−39(12)+101y = -39(12) + 101\newliney=−468+101y = -468 + 101\newliney=−367y = -367
  5. Find Second y-value: Find the y-value for the second x-value.\newlineFor x=−12x = -12:\newliney=−39(−12)+101y = -39(-12) + 101\newliney=468+101y = 468 + 101\newliney=569y = 569
  6. Write Coordinates: Write the coordinates in exact form.\newlineThe first coordinate is (12,−367)(12, -367).\newlineThe second coordinate is (−12,569)(-12, 569).

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