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Solve the system of equations.\newliney=−x+13y = -x + 13\newlinex2+y2=109x^2 + y^2 = 109\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Q. Solve the system of equations.\newliney=−x+13y = -x + 13\newlinex2+y2=109x^2 + y^2 = 109\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Substitute yy into second equation: Substitute yy from the first equation into the second equation.\newliney=−x+13y = -x + 13\newlinex2+y2=109x^2 + y^2 = 109\newlinex2+(−x+13)2=109x^2 + (-x + 13)^2 = 109
  2. Expand and simplify: Expand and simplify the equation.\newlinex2+(−x+13)2=109x^2 + (-x + 13)^2 = 109\newlinex2+(x2−26x+169)=109x^2 + (x^2 - 26x + 169) = 109\newline2x2−26x+169=1092x^2 - 26x + 169 = 109
  3. Move 109109 to left side: Move 109109 to the left side of the equation.\newline2x2−26x+169−109=02x^2 - 26x + 169 - 109 = 0\newline2x2−26x+60=02x^2 - 26x + 60 = 0
  4. Divide and simplify: Divide the equation by 22 to simplify.\newlinex2−13x+30=0x^2 - 13x + 30 = 0
  5. Factor the quadratic equation: Factor the quadratic equation.\newline(x−10)(x−3)=0(x - 10)(x - 3) = 0
  6. Solve for x: Solve for x by setting each factor equal to zero.\newlinex−10=0x - 10 = 0 or x−3=0x - 3 = 0\newlinex=10x = 10 or x=3x = 3
  7. Find corresponding y values: Find the corresponding y values for each x.\newlineFor x=10x = 10: y=−10+13=3y = -10 + 13 = 3\newlineFor x=3x = 3: y=−3+13=10y = -3 + 13 = 10
  8. Write coordinates in exact form: Write the coordinates in exact form.\newlineFirst Coordinate: (10,3)(10, 3)\newlineSecond Coordinate: (3,10)(3, 10)

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