I'm trying to prove that a solution exists to the following system of equations: $$ \frac{x}{b}=\frac{-e^y(2-e^x-e^y)}{(e^y-1)(e^y-e^x)} \\ \frac{y}{g}=\frac{-e^x(2-e^x-e^y)}{(e^x-1)(e^x-e^y)} $$ where $g>0>b$. I don't believe a solution can be expressed in closed form, but I would nevertheless like to show that it exists.
There's a possibility that Lambert's W function is involved, as one can divide the first line by the second, and get an expression which is solvable for $x$ using the Lambert function: $$x=\frac{bye^y}{g(1-e^y)}+\mathbb{W}(Z(y))$$ where $Z(y)=\frac{ybe^{y+\frac{bye^y}{g(e^y-1)}}}{g(e^y-1)}$.