How to show that there is at least one solution for the following equations ($x,y,z$ are unknowns)?
$$p_1 a_1 x + p_2 a_2 y + a_3 z = a_4 \; mod(p_1 p_2)$$
$$p_1 b_1 x + p_2 b_2 y + b_3 z = b_4 \; mod(p_1 p_2)$$
where $p_1$ and $p_2$ are two prime numbers. I want to find unknown values of $x,y,z$ based on known values of $\{a_1, a_2, a_3, a_4\}$ and $\{b_1, b_2, b_3, b_4\}$.