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.

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

so the possible values of x*y are 6 and 12

Posted by Jer on 2012-01-27 11:10:40