Numerical solution for transcendental equation in two variables

Question

I have this set of two equations

\begin{eqnarray} e^{-35 (y-x)} & = & \frac{x}{y} \tag{1} \\ \frac{e^{-90 x}-e^{-90 y}}{e^{-142 x}-e^{-142 y}} & = & \frac{1892}{1007} \tag{2} \end{eqnarray}

in the two real variables $x,y$, moreover I know from the Physical origin of these equations that $x,y\in[0,1]$. I am interested in numerical solutions, however Mathematica for example doesn't give a solution.

Any hints how I can proceed here?

Answer 1

Why do you say Mathematica does not give a numerical solution? Just use the command FindRoot[] taking initial approximations according to the graph shared by @MattiP. in the comments.This way you will be able to get approximations for all three solutions (one at a time).

Answer 2

From the first equation $$y=-\frac{1}{35} W\left(-35 e^{-35 x} x\right)$$ Plug in the second to have only a function of $x$.

Plot it and notice the root close to $x=0.055$. Using Newton method, the iterates are $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.05500000000 \\ 1 & 0.05459102858 \\ 2 & 0.05459414198 \\ 3 & 0.05459414216 \end{array} \right)$$ and $y=0.01252009470$.

You have a second solution swapping $x$ and $y$ (the symmetry being obvious even without @Matti P.'s nice plot).