That's the linear diophantine equation:
$Ax + By = c$ where $A,B,x,y,c \in Z$
We can represent A and B as below:
$GCD(A,B) = g$
$A = k_1*g$
$B= k_2*g$
So we can represent original equation like this:
$k_1*g*x + k_2 * g*y = c$
If we divide both sides by $g$, we get:
$k_1*x+k_2*y = c\div g$
So while $LHS \in Z$, then $RHS \in Z$ also if solution exists, so $c \mod g = 0$
Finally, the equation can be written as: $k_1*x + k_2*y = k_3*g$
Knowing this, can we be sure that solution will exists for every equation like
$A_1b_1 + A_2b_2 + A_3b_3 + ... + A_nb_n = c$
Can we? Any proof? Of course, our GCD will be $g = GCD(A_1,A_2,A_3,...,A_N)$ and $c \mod g = 0$