Proving a coprime certificate of (x,y)

Question

Let $a,b \in Z $ not both $0$. To verify $gcd(a,b) = d$ we can produce a certificate $x,y \in Z$ such that $ax + by = d$. Prove that any such certificate $(x,y)$ we produce must be coprime.

I know for coprime numbers $gcd(a,b) = 1$ where $a$ and $b$ represent the coprime numbers. I thought about doing a direct proof with an example but that doesn't prove this statement for any $(x,y)$. How else could I prove this?

Answer 1

$$ \gcd(a,b) = d $$ $$ ax+by = d $$ $$ \left( \frac{a}{d} \right) x + \left( \frac{b}{d} \right) y = 1 $$