Why is it as a consequence of the Euclidean Algorithm that $\displaystyle (a,b) = ax + by $?

I know we start off with $\displaystyle r_n = r_{n-2} - q_{n}r_{n-1} $. From here do we solve for $\displaystyle r_{n-2} $ and $\displaystyle r_{n-1} $ in terms of the previous remainders eventually solving for $\displaystyle r_n $ in terms of $\displaystyle a,b $?

Thanks