I'm looking for the solutions to $$\begin{eqnarray} z &=& x_0 + i \Delta_x \\ z &=& y_0 + j \Delta_y \\ i &\ge& 0\\ j &\ge& 0 \end{eqnarray}$$
Well, I know that
there is a solution only if $y_0 - x_0$ is a multiple of $GCD(\Delta_x, \Delta_y)$.
the formula for $z$ is $z = z_0 + k \Delta_z$ with $\Delta_z = LCM(\Delta_x, \Delta_y)$ and $k\ge 0$;
But I'm unable to compute a general formula fo $z_0$.
You'd be better off looking for an algorithm than an algebraic "formula".
First, suppose without loss of generality that $y_0>x_0$. Also, for simplicitly, we can assume that $x_0=0$ (otherwise, subtract $x_0$ from both $x_0$ and $y_0$ and add it to $z_0$ at the end).
Now if $gcd(\Delta_x,\Delta_y)>1$, divide everything by the gcd and multiply it in afterwards. Then we can assume that $\Delta_x$ and $\Delta_y$ are coprime.
What we're looking for is then the least $j\ge 0$ such that $$y_0+j\Delta_y \equiv 0 \pmod{\Delta_x} $$
But that must be the principal representative of the solution to $j_0\equiv -y_0\frac{1}{\Delta_y} \pmod{\Delta_x}$. You can compute the reciprocal of $\Delta_y$ modulo $\Delta_x$ using the extended Euclidean algorithm, and then reduce its product with $-y_0$ modulo $\Delta_x$ afterwards.
Finally compute $z_0 = y_0+j_0\Delta_y$, and then apply the corrections for the "without loss of generality" above.