Let me try to find the general solution for the quasi-linear P.D.E $$P(x,y,z)z_x+Q(x,y,z)z_y=R(x,y,z),~~~~~~~~~~~~~~~~~~(a)$$using Lagrange's method of characteristics (where $P,Q$ and $R$ are assumed to be $C^1$ functions in some open set $\Omega=D \times \Bbb R \subseteq\Bbb R^3$).
First point: The equation $(a)$ is of the form $$\langle (z_x(x,y),z_y(x,y),-1), (P(x,y,z(x,y)),Q(x,y,z(x,y)),R(x,y,z(x,y))) \rangle=0,$$
Second point: First point says that, the vector $(P,Q,R)$ is orthogonal to each normal $(z_x,z_y,-1)$ for the potential integral surface $z=z(x,y)$,
Third point : Second point says that, the vector $(P(x,y,z(x,y)),Q(x,y,z(x,y)),R(x,y,z(x,y)))$ is tangential to the surface $z=z(x,y)$ where I am not sure about the direction,
Forth point: Third point provides a parametric system of O.D.E $$\frac{dx}{dt}(t)=P(x(t),y(t),z(x(t),y(t))),$$ $$\frac{dy}{dt}(t)=Q(x(t),y(t),z(x(t),y(t))),$$ $$\frac{dz}{dt}(t)=R(x(t),y(t),z(x(t),y(t))),$$ so that $(x(t),y(t)) \in D$.
Fifth point: Solve the above system to deduce a general solution $$F(u,v)=0,$$ for $(a)$, where $F \in C^1(\Bbb R^2)$ and $u=u(x,y,z)=c_1,v=v(x,y,z)=c_2$ are independent solutions for the above system. Here, I am not sure about the elimination of the parameter $t$ and the extension of the characteristic curve in to the general integral surface.
I would appreciate the clearance of ambiguities indicated in second and fifth points,
thanks in advance.
I am not going to directly answer you question, but I am going to explain how do I interpret geometrically the characteristic method. I am only considering linear PDEs (homogeneous first, and then inhomogeneous) but I think it is enough to grasp the idea. I am changing to a notation which is easier for me.
Given a first order homogeneous linear PDE $$ a_{1} \partial W / \partial x_{1}+\cdots+a_{n} \partial W / \partial x_{n}=0 \tag{1} $$ where $a_k=a_k(x_1,\ldots,x_n)$, we look for solutions $$ W: \mathbb{R}^n \rightarrow \mathbb{R}. $$
We can consider a solution like "surfaces" $S$ given implicitly by $W=cte$. Their tangent planes are given by the kernel of $dW$
If we define the vector field $A=a_1\partial x_1+a_2\partial x_2+\ldots+a_n \partial x_n$, a function $W$ is solution of (1) if $A(W)=0$, i.e., the integral curves of $A$ are contained in the surfaces $W=cte$.
The ODE system $$ \dot{x}=A(x) \tag{2} $$ is called the characteristic system or characteristic equation, and their integral curves are called the characteristics of the original PDE. Solutions of the PDE are first integrals of the characteristic equation.
The ODE (2) is, for me, like a parametric expression of a line $r$ in linear algebra. On the contrary, equation (1) is similar to an implicit expression of all the planes containing the line $r$. The 1-form $dW$ is like an implicit equation.
Method of characteristics
I think that method of characteristics is usually called to what follows. The line $r$ above can be described by saying that it is made of vectors such that they are proportional to $(a_1,\ldots,a_n)$, that is, $$ \frac{dx_1}{a_1}=\frac{dx_2}{a_2}=\cdots $$ We can think of this expression as the symmetric form of a line in space.
From here, in the case of linear algebra, we recover several planes whose intersection gives the line $r$: $$ a_2 x_1-a_1 x_2=0 $$ $$ a_3 x_2-a_2 x_3=0 $$ $$ ... $$ In the differential context, we could recover an independent set of first integral if we were able to solve $$ a_2 \mbox{d}x_1-a_1 \mbox{d}x_2=0 $$ $$ a_3 \mbox{d}x_2-a_2 \mbox{d}x_3=0 $$ $$ ... $$ The solutions are independent first integrals of the ODE or functions that generate functionally the solutions of the PDE.
In general case $a_i$ can depend on too many variables, and the system cannot be solved, although I think it can be useful for numerical approximation of initial or boundary value problems.
Inhomogeneous case (your case)
If in the PDE we look for solutions in the form of explicit functions $$ W=x_n-f(x_1,\ldots,x_{n-1}) $$ we get $$ a_{1} \partial W / \partial x_{1}+\cdots+a_{n-1} \partial W / \partial x_{n-1}=a_n $$ which is the case for non-homogeneous linear PDEs.
DISCLAIMER
I am relatively new to this stuff, and what I have explained can be wrong. I you find mistakes please let me know before downvote.