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