How to determine the Steady State of this model? Consider the system of two equations: $$y_t=\beta\mathbb{E}_t[y_{t+1}+\gamma\cdot z_{t+1}]$$ $$x_t=\rho x_{t+1}+y_t$$ $$z_t=(z_0-Z_t)e^{-at}+z_T$$
  Determine the steady state.


The solution manual says: $$y=\frac{\beta\gamma z}{1-\beta}$$ $$x=\frac{\beta\gamma z}{(1-\beta)(1-\rho)}$$
Now I have no idea how to derive these values. The lecture sheets don't mention anything, except that the steady state is solved when we have the value of the shock $z_t$ (intial and terminal). Could anyone explain me the mathematical derivation of the steady state?
In general, a steady state in discrete time is that the first difference is zero at some $t=t_0$, and that it remains so: $$y_t-y_{t-1}=x_t-x_{t-1}=0$$ for all $t\geq t_0$.

Hence, in a steady state, $y_t$ and $x_t$ are constant at some values $y$ and $x$. Thus, $y=\beta(y+\gamma z)$, which implies $$y=\frac{\beta\gamma z}{1-\beta},$$ and $x=\rho x+y$, which implies $$x=\frac{y}{1-\rho}=\frac{\beta\gamma z}{(1-\rho)(1-\beta)}.$$