Proof that $ax+by+cz=0$ has infinitely many solutions.

Question

For all non-zero integers $x,y,z$ clearly there exist infinitely many non-zero integers $a,b,c$ such that $$ax+by+cz=0$$ How can I prove this simple statement?

Answer 1

We assume that at least one of $x,y,z$ is non-zero, without loss of generality $z\neq 0$.

First note there is at least one non-zero solution:

Let $a=-z,b=-z,c=x+y$. we get $-zx+-zy+(x+y)z$.

Now notice if $(a,b,c)$ works then $(ka,kb,kc)$ works.