Is this nonlinear system "solvable"?

Question

I don't really mean solvable, what I mean is: is it possible to rewrite $w$ in terms of $y$ and $z$? $$w =e^{3s}(1+r) -1$$ $$y = se^{2s}$$ $$z = 2se^{2s}(1+r) +r$$

I have fooled around with this for about an hour and the best thing I've got is that I can rewrite $r$ as a function of $y$ and $z$ only but I can't find one for $s$. Is there a "go to" method for these nonlinear systems or is it just a solve it any way you can type thing? Is there a way to know if it's even doable?

Answer 1

With $$s=1/2\,{\rm W} \left(2\,y\right)$$ (where W is the so called-LambertW-function) and $$r=\frac{z-2y}{2y+1}$$ you can eliminate $r,s$

Answer 2

An explicit form is not possible without the recourse to the Lambert function, as shown by Dr. Sonnhard Graubner.

Anyway, you can write the delicious implicit form below:

$$(w+1)^2(z+1)^2\left(\log(w+1)+\log(z+1)-\log(2y+1)\right)^3-27y^3(2y+1)^2=0.$$