If (x, y ) is a solution to the system of equations \(xy=6 \) and \(x^2y + xy^2 + x + y = 63\), find \(x^2 + y^2\)
xy = 6
x^2 y + xy^2 + x + y = 63 ⇒ xy ( x + y) + x + y = 63 ⇒ (x + y) (xy + 1) = 63
So
(x + y) ( 6 + 1) = 63
(x + y) * 7 = 63
x+ y = 63 / 7
x + y = 9 square both sides
x^2 + 2xy + y^2 = 81
x^2 + y^2 + 2(6) = 81
x^2 + y^2 + 12 = 81
x^2 + y^2 = 81 - 12
x^2 + y^2 = 69
