Form differential equation pair from discrete time system block diagram
Solving this via the z-transform is possibly the easiest route, however, proceeding with the difference equations:

$$\small v(k)=\frac{1}{2}v(k-1)+u(k)$$
$$\small y(k)=-\frac{1}{2}v(k)+v(k-1)$$

Form two simultaneous equations:

$$\small v(k)-\frac{1}{2}v(k-1)=u(k) $$
$$\small -\frac{1}{2}v(k)+v(k-1)=y(k) $$

Solving these for \$\small v(k-1)\$ gives:
$$\small v(k-1)=\frac{4}{3}y(k)+\frac{2}{3}u(k)\:\:\:...\:(1)$$

and solving these for \$\small v(k)\$ gives:
$$\small v(k)=\frac{2}{3}y(k)+\frac{4}{3}u(k)$$

Delaying the last equation by one time increment:
$$\small v(k-1)=\frac{2}{3}y(k-1)+\frac{4}{3}u(k-1)\:\:\:...\:(2)$$

Equating \$\small (1)\$ and \$\small (2)\$ results in:

$$y(k)=\frac{1}{2}y(k-1)+u(k-1)-\frac{1}{2}u(k) $$