I'm slightly confused on the definition of an equivalence class. Suppose $R$ is a relation on $Z \times (Z - {0})$ by $(a,b)R(c,d)$ if and only if $ad = bc$. What would a single equivalence class from this look like?
$(-3, 2)$ is related to $(-3, 2)$, and $(-3,-3)R(-3,-3)$ - are these in the same equivalence class?
The relation you describe is the one used to define the set of rationals $\Bbb{Q}$. Namely, an equivalence class $[a,b]_R$ for this relation can be represented by the fraction $\frac{a}{b}$ and for every fraction $\frac{c}{d}=\frac{a}{b} \in \Bbb{Q}$, you have $(c,d)\in [a,b]_R$.