The answer is that for all values of x. Notice that

in conjunction with and leads to

.

You can see that this is the only solution by what you derived:

have only one simultaneous solution: y = x = 0. So your solution is not incorrect, you simply didn't take it far enough. It is interesting to note that if you take any line y = mx you will get a second equation by your method: y = nx where m is not equal to m.

If it helps you to see what's going on here, let

which can be written as the system of equations

and

which only has one intersection point: (x, y, z) = (0, 0, 0).

-Dan