Generalizing a Pair of Diophantine Equations

For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\ =\ \frac{(a-1)(b-1)}{2} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by rep