Let's say that we have the second-order PDE which has hyperbolic form
\begin{align}u_{xx}+yu_{yy}=0 \label{1}\tag1 \end{align}
We know $a=1, \ b=0,\ c=y$
Thus, the discriminant is: $$d=b^2-ac=-y$$
So the equation is hyperbolic form when $y<0$.
It is clear that
\begin{align}u_{xx}+yu_{yy}&=0 \Leftrightarrow \\ \frac{\partial^2{u}}{\partial{x}^2}&=-y \frac{\partial^2{u}}{\partial{y}^2} \end{align}
The question is how to formulate the \eqref{1} as a first order PDE system: $$\frac{\partial{Q}}{\partial{x}}+A \frac{\partial{Q}}{\partial{y}}=0$$ where $Q$ vector in $\mathbb{R}$ and A is matrix $2 \times 2$ ?