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Solve the system of equations.\newliney=−27x+1y = -27x + 1\newliney=x2−26x−41y = x^2 - 26x - 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=−27x+1y = -27x + 1\newliney=x2−26x−41y = x^2 - 26x - 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=−27x+1y = -27x + 1\newliney=x2−26x−41y = x^2 - 26x - 41\newlineTo find the intersection points, we set the two equations equal to each other.\newline−27x+1=x2−26x−41-27x + 1 = x^2 - 26x - 41
  2. Rearrange and Form Quadratic Equation: Rearrange the equation to bring all terms to one side and set it equal to zero to form a standard quadratic equation.\newlinex2−26x−41+27x−1=0x^2 - 26x - 41 + 27x - 1 = 0\newlinex2+x−42=0x^2 + x - 42 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation.\newlineWe look for two numbers that multiply to −42-42 and add up to 11. These numbers are 77 and −6-6.\newline(x+7)(x−6)=0(x + 7)(x - 6) = 0
  4. Solve for x: Solve for x by setting each factor equal to zero.\newlinex+7=0x + 7 = 0 or x−6=0x - 6 = 0\newlinex=−7x = -7 or x=6x = 6
  5. Find yy for x=−7x = -7: Find the corresponding yy-values for each xx-value by substituting back into one of the original equations. We can use y=−27x+1y = -27x + 1.\newlineFor x=−7x = -7:\newliney=−27(−7)+1y = -27(-7) + 1\newliney=189+1y = 189 + 1\newliney=190y = 190
  6. Find yy for x=6x = 6: Find the corresponding yy-value for x=6x = 6:
    y=−27(6)+1y = -27(6) + 1
    y=−162+1y = -162 + 1
    y=−161y = -161
  7. Write Coordinates: Write the coordinates in exact form.\newlineThe first coordinate is (−7,190)(-7, 190).\newlineThe second coordinate is (6,−161)(6, -161).

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