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Solve the system of equations.\newliney=−29x−40y = -29x - 40\newliney=x2−17x−5y = x^2 - 17x - 5\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=−29x−40y = -29x - 40\newliney=x2−17x−5y = x^2 - 17x - 5\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=−29x−40y = -29x - 40\newliney=x2−17x−5y = x^2 - 17x - 5\newlineSet the two equations equal to each other to find the xx-values where they intersect.\newline−29x−40=x2−17x−5-29x - 40 = x^2 - 17x - 5
  2. Rearrange to Standard Form: Rearrange the equation to get a standard form quadratic equation.\newlinex2−17x−5+29x+40=0x^2 - 17x - 5 + 29x + 40 = 0\newlinex2+12x+35=0x^2 + 12x + 35 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation.\newlineIn 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.\newlinex2+12x+35=0x^2 + 12x + 35 = 0\newline(x+7)(x+5)=0(x + 7)(x + 5) = 0
  4. Solve for x: Solve for x.\newlineSet each factor equal to zero, and solve for x.\newline(x+7)=0(x + 7) = 0 or (x+5)=0(x + 5) = 0\newlinex=−7x = -7 or x=−5x = -5
  5. Find yy for x=−7x=-7: Find the corresponding yy-values for each xx-value by substituting back into either of the original equations. We'll use y=−29x−40y = -29x - 40. For x=−7x = -7: y=−29(−7)−40y = -29(-7) - 40 y=203−40y = 203 - 40 y=163y = 163
  6. Find yy for x=−5x=-5: Find the corresponding yy-value for x=−5x = -5:
    y=−29(−5)−40y = -29(-5) - 40
    y=145−40y = 145 - 40
    y=105y = 105
  7. Write Coordinates: Write the coordinates in exact form.\newlineFirst Coordinate: (−7,163)(-7, 163)\newlineSecond Coordinate: (−5,105)(-5, 105)

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