Looks like all you need is x = x(u,v), y = y(u,v).

Well try multiplying your two expressions together. That is,

u * v = (xy) * (y/x) = y^{2}.

Similarly, dividing gives

u / v = (xy) / (y/x) = x^{2}.

Now you have,

x = (u/v)^{1/2}, y = (u*v)^{1/2}.

Hope that's enough to get you on the right track. Let me know otherwise and I'll try and clear it up.