Is there a short proof for Bézout's lemma when $\gcd(a,b)=1$?

Question

Bézout's lemma states that there exist integers $x$ and $y$ such that

$$ax+by=\gcd(a,b)$$

Is there some short proof for this when $a$ and $b$ are coprime? As opposed to something like this, like, from an algebraic/ring theoretic perspective.