The Chain Rule & Implicit Differentiation · Challenge problem

Let $z(x,y)$ be a smooth function, and introduce new variables $u = x + y$, $v = x - y$. The chain rule turns $z_{xx} - z_{yy}$ into a constant multiple of $z_{uv}$: $z_{xx} - z_{yy} = k\,z_{uv}$ for every such $z$. Find $k$.