Problem. Let $I=(x-y)$ and $J=(x+y)$. Describe $\mathbb{C}[x,y]/I$, $\mathbb{C}[x,y]/J$, $\mathbb{C}[x,y]/(I+J)$, $\mathbb{C}[x,y]/IJ$ as polynomial functions on a certain subset in $\mathbb{C}^2$.
I was given the hint to make the change of variables $x'=x+y$ and $y'=x-y$, but I am having difficulties describing these.