Proof

Suppose that xa + yb = 0 and that gcd(a,b) = 1. From xa = -by it follows that a | by. Since gcd(a,b) = 1, we find a | y, see the proposition of Section 1.2. But that means that there exists an integer n such that an = y. Substitution of y in the original equation gives x = -nb . This proves the lemma.