Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$.

Question

So i know the question has been answered already but the answer uses isomorphisms that I have not been taught. I was wondering if you can show the factor group is a field directly. Let us assume that $A$ is in fact an ideal. Then we look at the factor group $(Z \oplus Z)/A$ and try to show it is a field. Let us take an arbitrary element of the factor group $(a,b)+A$. We want to show we have a multiplicative inverse so $(a,b)*(c,d)+A=(1,1)+A$ but I don't see a solution here. Any help would be appreciated. Thank you