The equations are simple.
You will know them.

What you can't calculate, but have to understand (so no equations), is that you have to use the distance between exit pupil and film, instead of the focal length.
You don't appear ready to accept that, but you could of course also ask Emmanuel. I'm sure he will explain.

Re: "translation of axes". It works perfectly fine. But you have to remember that when you flip a lens, the entrance pupil turns into the exit pupil, the exit pupil into the entrance pupil. And with that, the f-stops of the lens change too (remember that, they are focal length/diameter entrance pupil.)
Maybe that's where your results go wrong?
(Maybe too that i don't get what you mean when you say "translation of axes"?)