I need to solve for variables $u$ and $v$ in this system of equations:
How do I isolate $u$ and $v$ to get them both in terms of $x$, $y$, and $k$?
If you expand the squares in the first equation, you can use the second to eliminate the $u^2,v^2$ terms. That leaves you with one linear equation and one quadratic. Solve the first for $u$ and substitute into the second. That gives you a quadratic in $v$ which you can solve, getting two roots. Plug them into the first and get two solutions for $u$. Check them both and you are done.