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Solve the system of equations.\newliney=x2−14x−23y = x^2 - 14x - 23\newliney=−14x+41y = -14x + 41\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=x2−14x−23y = x^2 - 14x - 23\newliney=−14x+41y = -14x + 41\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=x2−14x−23y = x^2 - 14x - 23\newliney=−14x+41y = -14x + 41\newlineSet the two equations equal to each other to find the xx-values where they intersect.\newlinex2−14x−23=−14x+41x^2 - 14x - 23 = -14x + 41
  2. Simplify and Rearrange: Simplify the equation by adding 14x14x to both sides and subtracting 4141 from both sides to get the quadratic equation in standard form.\newlinex2−14x−23+14x−41=−14x+41+14x−41x^2 - 14x - 23 + 14x - 41 = -14x + 41 + 14x - 41\newlinex2−64=0x^2 - 64 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation to find the values of xx.x2−64=(x−8)(x+8)x^2 - 64 = (x - 8)(x + 8)Set each factor equal to zero and solve for xx.(x−8)=0(x - 8) = 0 or (x+8)=0(x + 8) = 0x=8x = 8 or x=−8x = -8
  4. Solve for x: Find the corresponding yy-values for each xx-value by substituting back into one of the original equations. We'll use y=−14x+41y = -14x + 41. For x=8x = 8: y=−14(8)+41y = -14(8) + 41 y=−112+41y = -112 + 41 y=−71y = -71
  5. Find yy for x=8x=8: Find the corresponding yy-value for the second xx-value.\newlineFor x=−8x = -8:\newliney=−14(−8)+41y = -14(-8) + 41\newliney=112+41y = 112 + 41\newliney=153y = 153
  6. Find yy for x=−8x=-8: Write the coordinates in exact form.\newlineThe first coordinate is (8,−71)(8, -71).\newlineThe second coordinate is (−8,153)(-8, 153).

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