I have solved the following diophantine equations:
- $14x+35y=93$
- $56x+72y=40$
That's what I have tried:
- $gcd(35,14)=7$ , but $7 \nmid 93,$ so the first diophantine equation has no solution.
- $gcd(56,72)=8$, $8 \mid 40$,so the diophantine equation $56x+72y=40$ has infinite solutions. We are looking for $x_0,y_0$ such that $56x_0+72y_0=8$.This relation is satisfied for $x_0=4,y_0=-3$.
So, $x_1=20, y=-15$ is a solution of $56x+72y=40$.
Therefore, the solutions of this diophantine equation are given by the formulas:
$$x=20+7k, y=-15-9k, k \in \mathbb{Z}$$
Could you tell me if that I have tried is right?