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Solve the system of equations.\newliney=−13x+39y = -13x + 39\newliney=x2−19x−1y = x^2 - 19x - 1\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=−13x+39y = -13x + 39\newliney=x2−19x−1y = x^2 - 19x - 1\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the following system of equations:\newliney=−13x+39y = -13x + 39\newliney=x2−19x−1y = x^2 - 19x - 1\newlineTo find the solution, we need to set the two equations equal to each other because they both equal yy.\newline−13x+39=x2−19x−1-13x + 39 = x^2 - 19x - 1
  2. Rearrange to Standard Form: Rearrange the equation to get a standard form of a quadratic equation by moving all terms to one side.\newlinex2−19x−1+13x−39=0x^2 - 19x - 1 + 13x - 39 = 0\newlinex2−6x−40=0x^2 - 6x - 40 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation to find the values of xx. We are looking for two numbers that multiply to −40-40 and add up to −6-6. These numbers are −10-10 and 44. (x−10)(x+4)=0(x - 10)(x + 4) = 0
  4. Solve for x: Solve for x by setting each factor equal to zero.\newlinex−10=0x - 10 = 0 or x+4=0x + 4 = 0\newlineThis gives us two solutions for x:\newlinex=10x = 10 or x=−4x = -4
  5. Find Corresponding y-Values: Find the corresponding y-values for each x-value by substituting back into one of the original equations. We can use y=−13x+39y = -13x + 39.\newlineFor x=10x = 10:\newliney=−13(10)+39y = -13(10) + 39\newliney=−130+39y = -130 + 39\newliney=−91y = -91\newlineFor x=−4x = -4:\newliney=−13(−4)+39y = -13(-4) + 39\newliney=52+39y = 52 + 39\newliney=91y = 91
  6. Write Coordinates: Write the coordinates in exact form.\newlineThe solutions to the system of equations are the points where the two graphs intersect, which are the xx-values we found and their corresponding yy-values.\newlineFirst Coordinate: (10,−91)(10, -91)\newlineSecond Coordinate: (−4,91)(-4, 91)

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