Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$
Solution 1:
Your proof regarding the fact that $R$ is in fact an equivalence relation is indeed correct.
Now think what the equivalence class of $(1,1)$ could be, since $(1,1) R (1,1)$ you know that everyone in the equivalence class has the property that $(a,b)$ holds $a+1=b+1$, therefore $a=b$.
This means that the equivalence class of $(1,1)$ is $(a,a)$ for $a\in\mathbb{N}$.
Solution 2:
HINT: The equivalence class of $(1,1)$ is made up of all pairs $(x,y)\sim(1,1)$. Write explicitely what the latter means and get a relation that need to be satisfied by $x$ and $y$.