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Solve the system of equations.\newliney=5x+34y = 5x + 34\newliney=13x2+31x+47y = 13x^2 + 31x + 47\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)

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Q. Solve the system of equations.\newliney=5x+34y = 5x + 34\newliney=13x2+31x+47y = 13x^2 + 31x + 47\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=5x+34y = 5x + 34\newliney=13x2+31x+47y = 13x^2 + 31x + 47\newlineTo find the intersection points, we set the two equations equal to each other.\newline5x+34=13x2+31x+475x + 34 = 13x^2 + 31x + 47
  2. Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation.\newline0=13x2+31x+47−5x−340 = 13x^2 + 31x + 47 - 5x - 34\newline0=13x2+26x+130 = 13x^2 + 26x + 13
  3. Simplify by Dividing: Notice that all coefficients are divisible by 1313, so we can simplify the equation by dividing by 1313.0=x2+2x+10 = x^2 + 2x + 1This is a perfect square trinomial.
  4. Factor Perfect Square Trinomial: Factor the perfect square trinomial.\newline0=(x+1)(x+1)0 = (x + 1)(x + 1)\newline0=(x+1)20 = (x + 1)^2
  5. Solve for x: Solve for x.\newlineSet the factor equal to zero and solve for x.\newline(x+1)=0(x + 1) = 0\newlinex=−1x = -1
  6. Find y-Value: Find the corresponding y-value by substituting x=−1x = -1 into either of the original equations. We'll use the first equation for simplicity.y=5(−1)+34y = 5(-1) + 34y=−5+34y = -5 + 34y=29y = 29
  7. Write Coordinates: Write the coordinates in exact form.\newlineSince the quadratic equation had a repeated root, there is only one intersection point.\newlineThe coordinate is (−1,29)(-1, 29).

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