Application of Bezout's theorem

Question

Let we have $p_1q_2 -p_2q_1 = \pm 1$ where $p_1,p_2,q_1,q_2 \in \mathbb{Z}$ and let $\cfrac{a}{b} \in \mathbb{Q}$. Then, how can we show that there are integers $x,y$ such that $$xp_1+yp_2=a, xq_1+yq_2=b.$$

Using linear algebra we can deduce the result by putting $p_i,q_j$'s inside a $2\times 2$ matrix. Is there any other way to prove it?

Answer 1

So, $\gcd(p_1,p_2)=1$, hence the set $\{p_1x+p_2y\mid x,y \in\mathbb{Z}\} =\mathbb{Z}$. So there is $x,y \in \mathbb{Z}$ such that $p_1x+p_2y=a$. Others goes similar ways