Without solving for x and y, determine all possible values of x*y in this simultaneous equation:

x+x*y = 8 and, y+x*y = 9

Note: Solving for 1/x, 1/y, -x or -y is similarly NOT allowed.

Add the two equations together to get:
(x+y) + 2xy = 17
x+y = 17 - 2xy

Also multiply the two equations together to get:
(x+xy) * (y+xy) = 72
xy + (x+y)*xy + (xy)^2 = 72

Substitute my first equation into the second to get:
xy + (17 - 2xy)*xy + (xy)^2 = 72
18xy - (xy)^2 = 72
(xy)^2 - 18xy + 72 = 0
(xy - 6) * (xy - 12) = 0
xy = 6 or 12

Posted by Brian Smith on 2012-01-28 14:21:27