Simplfiying a system output equation
These problems are most easily solved by using the $\mathcal{Z}$-transform. But it is also possible to do it in the sample domain. You have two equations for expressing $w[n]$ and $w[n-1]$ in terms of $x[n]$ and $y[n]$:

$$x[n]=w[n]-aw[n-1]\tag{1}$$
$$y[n]=w[n]+bw[n-1]\tag{2}$$

From $(1)$ and $(2)$, $w[n]$ is obtained as

$$w[n]=\frac{1}{a+b}\big(bx[n]+ay[n]\big)\tag{3}$$

Plugging $(3)$ into $(2)$ gives

$$y[n]=\frac{1}{a+b}\big(bx[n]+ay[n]+b^2x[n-1]+aby[n-1]\big)\tag{4}$$

Bringing all terms with $y[n]$ to the left side gives

$$y[n]\frac{b}{a+b}=\frac{1}{a+b}\big(bx[n]+b^2x[n-1]+aby[n-1]\big)\tag{5}$$

which finally results in

$$y[n]=x[n]+bx[n-1]+ay[n-1]\tag{6}$$