Show that there are infinitely many integral solutions

Question

How do I show that there infinitely man integral solutions of an equation?

For example

$1947*m+264*n=33$

I used the Euclidean algorithm and then back substituted to find one pair of solution $m=3, n=-22$. But I'm not sure how to find more or prove it.

My textbook hints that - these coefficients are not necessarily produced by euclidean algorithm

Answer 1

HINT.-If you have a solution of the equation $ax+by=c$ say $ax_0+by_0=c$ then you have

$$a(x-x_0)+b(y-y_0)=0$$ hence, making $x-x_0=bt$ and $y-y_0=-at$, you have the general solution $$(x,y)=(x_0+bt,\space y_0-at)$$ where $t$ is a parameter.

You have, for instance,$(x,y)=(-13,96)$ (note first that equation reduces to $59x+8y=1$)