Let xy = a so also y = a/x Substitute into the first equation gives x + a = 8 then rewrite this as x = 8 - a Substitute into the second equation gives a/x + a = 9 then solve for x gives x = a/(9-a)

Substitute now for each x: 8 - a = a/(9-a) 72 - 17a + a^2 = a a^2 - 18a + 72 = 0 this quadratic has solutions a= 6 or a=12