How do I solve this system of 2 equations? I need to solve for variables $u$ and $v$ in this system of equations:
$(x+u)^2+(y+v)^2=1$
$u^2+v^2=k$
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.
