The point (x,y) lies on both conics x2+xy+x=81 and y2+xy+y=51. Given that x+y is positive, determine the value of x+y.

Answer 1

x^2 + xy + x = 81
xy + y^2 + y = 51

add them to get

x^2 + 2xy + y^2 + x + y = 132
(x+y)^2 + (x+y) - 132 = 0
((x+y)+13)((x+y)-12) = 0
so, x+y = 12