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Solve the system of equations.\newliney=17x+22y = 17x + 22\newliney=x2+15x−2y = x^2 + 15x - 2\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=17x+22y = 17x + 22\newliney=x2+15x−2y = x^2 + 15x - 2\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=17x+22y = 17x + 22\newliney=x2+15x−2y = x^2 + 15x - 2\newlineSet the two equations equal to each other to find the xx-values where they intersect.\newline17x+22=x2+15x−217x + 22 = x^2 + 15x - 2
  2. Rearrange and Identify: Rearrange the equation to set it to zero and identify the standard form of the quadratic equation. \newlinex2+15x−2−17x−22=0x^2 + 15x - 2 - 17x - 22 = 0\newlinex2−2x−24=0x^2 - 2x - 24 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation to find the values of xx. In the quadratic equation ax2+bx+cax^2 + bx + c, the factors are of the form (x+m)(x+n)(x + m)(x + n), where bb is the sum and cc is the product of mm and nn respectively. x2−2x−24=(x−6)(x+4)x^2 - 2x - 24 = (x - 6)(x + 4)
  4. Solve for x: Solve for x by setting each factor equal to zero.\newline(x−6)=0(x - 6) = 0 or (x+4)=0(x + 4) = 0\newlinex=6x = 6 or x=−4x = -4
  5. Find y-Values: Find the corresponding y-values for each x-value by substituting back into one of the original equations.\newlineFor x=6x = 6, substitute into y=17x+22y = 17x + 22:\newliney=17(6)+22y = 17(6) + 22\newliney=102+22y = 102 + 22\newliney=124y = 124\newlineFor x=−4x = -4, substitute into y=17x+22y = 17x + 22:\newliney=17(−4)+22y = 17(-4) + 22\newliney=−68+22y = -68 + 22\newliney=−46y = -46
  6. Write Coordinates: Write the coordinates in exact form.\newlineThe first coordinate is (6,124)(6, 124).\newlineThe second coordinate is (−4,−46)(-4, -46).

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