Finding all solutions to $3x + 4y \equiv 1 \pmod 7$ Find all solutions $\pmod 7$: $3x + 4y$ is congruent to $1 \pmod 7$.
I have tried writing out the various equations such as $3x + 4y = 1$, $3x + 4y = 8$, $3x + 4y = 15$, etc., but I do not know how to find the finite solution.

Hint: for any $x,y $ we always have $3x+4y=3 (x+4)+4 (y-3) $.
So by induction:
$3x+4y\equiv 1 \mod 7 \implies 3 (x\pm 4k)+4 (y\mp 3k)\equiv 1 \mod 7$
So... if you know one solution, you know infinitely many.
The question is are there any not generated from your first solution? (I.e. can we make that $\implies $ into $\iff $?)
