Effect of pullback of differential forms on an ideal

Say that the exterior differential system (EDS) corresponding to a PDE system is:

$$df-f_x\,dx-f_y\,dy-f_w\,dw-f_z\,dz=0,\\ a_1\,f_x+a_2\,f_y=0,\tag{sys}$$

Of course we also require the independence condition, $dx\wedge dy\wedge dw\wedge dz\neq 0$.

1. Instead of (sys) can I simply use the following? $$ df +\dfrac{a_2}{a_1}f_y\,dx-f_y\,dy-f_w\,dw-f_z\,dz=0 \tag{sys$^\prime$}$$
2. I guess what I'm asking above is whether the ideal generated by $$\theta=df +\dfrac{a_2}{a_1}f_y\,dx-f_y\,dy-f_w\,dw-f_z\,dz$$ coincides with the pull-back of the ideal generated by the contact form ($i.e.$ the left-hand side of first line in sys) to the manifold in jet space given by the second line of sys?

I think the answer is yes because the pullback to the manifold in jet space defined by $a_1\,f_x+a_2\,f_y=0$ commutes with the exterior product and with the exterior derivative so the ideal generated by $\mathrm{sys^\prime}$ will coincide with the pullback of the ideal generated by $\mathrm{sys}$ but not 100% sure.

Sorry if all of this obvious but I'm not a mathematician. Besides Bryant et. al. I'm using "Cartan for Beginners" and "Lie's Structural Approach to PDE Systems". I would be grateful for any other useful references.

As you suspect the answer is yes, and you basically explain why it is so. If you are not convinced you could add the following details: your PDE $a_1 f_x + a_2 f_y=0$ is first order with the variable $f$ the dependent variable while $x,y,w,z$ are the independent variables. Standard coordinates in first jet space are $x,y,w,z,f,f_x,f_y,f_w,f_z$ while the contact form is what you say it is. What you are basically doing is restricting the coordinates $x,y,w,z,f,f_y,f_w,f_z$ (without $f_x$) to the submanifold determined by the PDE and using these as coordinates on the submanifold (maybe you want to check that these are indeed valid coordinates?). Then the pullback of the coordinate $f_x$ is given by $-\frac{a_1}{a_2}f_y$ and the pullback of the contact form is what you wrote by the reasons you gave :)

Edit: in addition to the two nice books you mentioned I'd also recommend the book "Symmetries and conservation laws for differential eqauations in mathematical physics"