General Equilibrium - Mas Colell Problem 17.D.2
Your answer is essentially the same as Mas-Colell's.

$z_{11}$ is excess demand for good $1$ by consumer $1$. $z_{12}$ is excess demand for good $1$ by consumer $2$. They sum up to zero at the price vector $(1,1)$, which means that the market for good $1$ clears. By Walras' law, that implies that the market for good $2$ clears as well, hence $(1,1)$ is an equilibrium price vector.