Joint entropy of sum and difference of random variables

Question

Let $X$~$U\left\{1,8 \right\}$ , $Y$~ $\begin{pmatrix} 1 & 2 & 3 & \dots \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \dots \end{pmatrix}$. I need to calculate $H(X+Y,X-Y)$. I can do it by computing probability distribution of $X+Y$ and $X-Y$, but I think, there is some other, more elegant way to do this. Thanks!