Given a set and an equivalence relation find the equivalence classes and the quotient set Given $A=\{1,2,3,4,5\}$ in $A\times A$ we define $$(x; y)R(z; t)\quad\Leftrightarrow\quad x + t = y + z.$$ Find the equivalence classes and the quotient set.

I tried to find the equivalence classes:
Let $(x_0,y_0)\in A\times A$. The class to which it belongs $(x_0,y_0)$ is $$C[(x_0,y_0)]=\{(x,y)\in A\times A:(x,y)R(x_0,y_0)\}=\{(x,y)\in A\times A:x+y_0=y+x_0\}.$$
Is it all right up here? Because then I do not know how to continue. Maybe passing the generic elements $ x_0, y_0 $ to a member? If I do not know the equivalence classes, I can not know the whole quotient. Some help?
Thank you!

Hint: rewrite the condition $x+t=y+z$ as $$x-y=z-t$$
The value of $x-y$ can be anything between $-4$ and $4$, so there are $9$ equivalence classes.

Geometrical interpretation: an equivalence class is a set of points all lying on the same line with slope $1$.
