A continued fraction of a inhomogeneous linear form

Let $\alpha$, $\beta$ be real numbers $0\le\alpha<1$, $0\le\beta<1$. They define at the plane $(y,z)\in\mathbb R^2$ the inhomogeneous linear form $L_{\alpha,\beta}(y,z)=-\beta+\alpha y+z$. We propose the algorithm of an expansion of this linear form into the ‘inhomogeneous continued fraction’
$$ L_{\alpha,\beta}\sim[0;b_1,b_2,\dots]\,\mathrm{mod}\,[0;a_1,a_2,\dots]. $$
Inhomogeneous continued fraction generalize the classic regular continued fraction: for $\beta=0$ every $b_n=0$ and we get the continued fraction expansion of the number $\alpha$: $L_{\alpha,0}\sim[0]\,\mathrm{mod}\,[0;a_1,a_2,\dots]$. Some properties of inhomogeneous continued fractions are proved.