In the proof of Lagrange's best approximations law it is stated that the system of equations $$ xp_n+yp_{n+1}=p\\ xq_n+yq_{n+1}=q\\ $$ where $\dfrac{p_n}{q_n}$ is the rational is a convergent of the continued fraction of a real number $\alpha$, has integer solutions $x, y$ since $p_nq_{n+1}-p_{n+1}q_n=(-1)^n$.
Please check CONTINUED FRACTIONS BEST APPROXIMATIONS
I understand that the system of equations can be written as $$ \begin{bmatrix}p_n&p_{n+1}\\q_n&q_{n+1}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}p\\q\end{bmatrix} $$ which has a solution if $\begin{vmatrix}p_n&p_{n+1}\\q_n&q_{n+1}\end{vmatrix}=p_nq_{n+1}-p_{n+1}q_n\neq 0$
How do we know that this system has integer solutions ?