So now we just have to solve for the final velocities 
and 
and we're done. But how do you do that? One equation is linear and
the other quadratic? If you try to substitute one into the other,
you get a big mess. You can get around this by being a little clever,
but there is a better way to think about all this which gives you
more physical insight into what's going on with the physics of this
problem.