I can show that this is an equivalence relation given that $b \ne 0$.
My question: Is it true that this relation is NOT an equivalence relation if we don't have the condition $b \ne 0$, because if $b=0$, then that would lead to division by 0, which is undefined?
You are trying to adjoin "infinity' to the set of rational numbers by introducing the class of $[(1,0)]$, and also you are trying to define $0/0$ by introducing $[(0,0)]$. The problem is that, by definition $(0,0)$ is related to all (a,b), therefor the transitivity will be violated.
For example you will have $(3,5)R (0,0)$ and $(0,0)R(4,5)$ without $(3,5)R(4,5)$