[SOLVED] Planes of some sort?

So in my calculus and vectors course, I was assigned the following question:

Determine ALL real values of x, y and z, that satisfy the following

system of equations:

1)x^2 - yz + xy + zx = 82

2)y^2 - zx + xy + yz = -18

3)z^2 - xy + zx + yz = 18

We have been learning about planes, so I am pretty sure I have to apply what I have learned about planes to this problem. But there are so many variables, and whatever I do (I've tried elimination and rearranging things) seems to lead me back to one of these equations! please help :)