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Solve the system of equations.\newliney=−23x+13y = -23x + 13\newliney=x2−13x−26y = x^2 - 13x - 26\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=−23x+13y = -23x + 13\newliney=x2−13x−26y = x^2 - 13x - 26\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=−23x+13y = -23x + 13\newliney=x2−13x−26y = x^2 - 13x - 26\newlineTo find the solution, we will set the two equations equal to each other since they both equal yy.\newline−23x+13=x2−13x−26-23x + 13 = x^2 - 13x - 26
  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.\newlinex2−13x−26+23x−13=0x^2 - 13x - 26 + 23x - 13 = 0\newlinex2+10x−39=0x^2 + 10x - 39 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation to find the values of xx. We are looking for two numbers that multiply to −39-39 and add up to 1010. These numbers are 1313 and −3-3. (x+13)(x−3)=0(x + 13)(x - 3) = 0
  4. Solve for x: Solve for x by setting each factor equal to zero.\newlinex+13=0x + 13 = 0 or x−3=0x - 3 = 0\newlinex=−13x = -13 or x=3x = 3
  5. Find y-values: Find the corresponding y-values for each x-value by substituting back into one of the original equations. We can use y=−23x+13y = -23x + 13.\newlineFor x=−13x = -13:\newliney=−23(−13)+13y = -23(-13) + 13\newliney=299+13y = 299 + 13\newliney=312y = 312
  6. Find y-value for x=33: Find the y-value for x=3x = 3:y=−23(3)+13y = -23(3) + 13y=−69+13y = -69 + 13y=−56y = -56
  7. Write Coordinates: Write the coordinates in exact form.\newlineThe solutions to the system of equations are the points where the two equations intersect.\newlineFirst Coordinate: (−13,312)(-13, 312)\newlineSecond Coordinate: (3,−56)(3, -56)

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