There are more roots than that. Observe: (remove the spaces, it's to get around the "no links" thing)
Go to wolframalpha .com / input /
But your two values are correct. There are actually 12 solutions — 6 x solutions, and 6 y solutions.
By the transitive property, you could set the equations equal to each other and try to simplify it down, then try to solve it by substitution using one of the original equations (probably the second equation). Don't know if that would help anything though, I haven't done anything like this, I'm just speculating.