$F(p,q,u,x,y)=0$ and $\frac{\partial u}{\partial x}=p$ and $\frac{\partial u}{\partial y}=q$
$$\frac{\partial F}{\partial p} \frac{\partial p}{\partial x} +\frac{\partial F}{\partial q} \frac{\partial q}{\partial x} +\frac{\partial F}{\partial u} \frac{\partial u}{\partial x} +\frac{\partial F}{\partial x}=0$$ Which implies $$\frac{\partial F}{\partial p} \frac{\partial p}{\partial x} +\frac{\partial F}{\partial q} \frac{\partial q}{\partial x} =-p\frac{\partial F}{\partial u}-\frac{\partial F}{\partial x}$$