In $R=\mathbb{C}[x,y]$: why is $(x,y-1)+(x-1,y)=R$?

Question

I have these two ideals in $\mathbb{C}[x,y]$: $I=(x,y-1)$ and $J=(x-1,y)$. I would like to show that there exist $i,j$ belonging to $I$ and $J$ respectively such that $i+j=1$, so that $I + J= \mathbb{C}[x,y]$

Answer 1

By definition, $I+J = \{i+j\mid i\in I, j\in J\}$. In the case given, take $i=x$ and $j=(-1)(x-1)=1-x$. Then $i+j=1$.