Solutions for exponential nonlinear system

[Mojo]

It's been a while since I've done math, perhaps someone can point me in the right direction. I'm trying to figure out ways to solve the following two systems.

e0 = e^(Ax + BxX0) + e^(Ay + ByY0)
e1 = e^(Ax + BxX1) + e^(Ay + ByY1)
e2 = e^(Ax + BxX2) + e^(Ay + ByY2)
e3 = e^(Ax + BxX3) + e^(Ay + ByY3)

e0 = e^(Ax + BxX0) + e^(Ay + ByY0) + e^(Az + BzZ0)
e1 = e^(Ax + BxX1) + e^(Ay + ByY1) + e^(Az + BzZ1)
e2 = e^(Ax + BxX2) + e^(Ay + ByY2) + e^(Az + BzZ2)
e3 = e^(Ax + BxX3) + e^(Ay + ByY3) + e^(Az + BzZ3)
e4 = e^(Ax + BxX4) + e^(Ay + ByY4) + e^(Az + BzZ4)
e5 = e^(Ax + BxX5) + e^(Ay + ByY5) + e^(Az + BzZ5)
e6 = e^(Ax + BxX6) + e^(Ay + ByY6) + e^(Az + BzZ6)

They are two different systems (not related), but in each system e, X, Y, and Z are known.

[Abandon]

posting for a very particular reason. If you can't figure it out, fuck you.

[BaneTheBrawler]

is it because you can no longer see the cinnamon sugar swirls in every bite?

[Mojo]

I have a theory on what it is.

In regards to the questions, they can both be solved using numerical approaches. I haven't been able to think of any other way yet.

[Quicklet]

What are you trying to solve for?

[Byrthnoth]

He wants to solve for Ax, Ay, Az, Bx, By, and Bz.

I took the nearest sharpie to the nearest paper and could only solve for two of the variables for the first equation system in a page.

(ln(e-e^(Ay + By*Y2)) - ln((e-e^(Ay+By*Y1))/X1))*X1/(X1-1) = Ax

Good luck solving the next two substitutions. Also, knowing how long it has been since I last did math, there's a high likelihood that I already made at least one algebra mistake.