This appears in the proof of Charpit's method for solving nonlinear first order PDEs:
Consider the two equaions: $$f(x,y,z,p,q)=0,\qquad \qquad (1)$$ $$g(x,y,z,p,q)=0,\qquad \qquad (2)$$ where $f$ and $g$ are both smooth. In Ayers' book "Differential equations", chapter 30, it is stated that:
for the equations (1) and (2) to be solvable for $p$ and $q$ we must have $$\begin{vmatrix} \frac{\partial f}{\partial p} & \frac{\partial f}{\partial q} \\ \frac{\partial g}{\partial p} & \frac{\partial g}{\partial q} \notag \end{vmatrix} \neq 0.$$
This is clear when both $f$ and $g$ are linear in both variables $p$ and $q$. What about the general case ?
Thanks a lot