How can we express x if we have
$$ x = (y - x) e^z$$
From $\displaystyle x= (y- x)e^z$, first do the indicated multiplication- $\displaystyle x= ye^z- xe^z$.
Now add $\displaystyle xe^z$ to both sides- $\displaystyle x+ xe^z= ye^z$.
Factor out x- $\displaystyle x(1+ e^z)= ye^z$.
Finally, divide both sides by $\displaystyle 1+ e^z$-
$\displaystyle x= \frac{ye^z}{1+ e^z}$.
With one caveat:
$x = \begin{cases}x \in \mathbb{C} & z = \pi i(2n+1), n\in \mathbb{Z} \\ \dfrac{ye^z}{1+e^z} & \text{otherwise}\end{cases}$
In other words, if $e^z = -1$, then $y=0$ and $x$ can be any complex number. Otherwise, it equals the fraction Plato and HallsofIvy gave.