$$13x + 12 = 22y + 19 = 20z + 3$$
I am able to solve the first diophantine equation $13x + 12 = 22y + 19$ in the classic form of $ax + by = c$ like $$13x_1 - 22y_1 = 7.$$ In this case the condition of existing solution is met: $gcd(a, b) | c \rightarrow gcd(13, -22) | 7$. The solution is $x_1 = -35 + 22n, y_1 = -21 + 13n$.
I am also able to solve the equations:
- $13x_2 + 12 = 20z_1 + 3$. The solution is $x_2 = 27 + 20n, z_1 = 18 + 13n$.
- $22y_2 + 19 = 20z_2 + 3$. The solution is $y_2 = -8 + 10n, z_2 = -8 + 11n$.
Based on that, it is possible to "merge" solutions $x_1 = -35 + 22n$ and $x_2 = 27 + 20n$ into $x$ that fullfils the original equation?