Price elasticity and optimal pricing in a monopoly with zero marginal cost
Hint

You are probably working with (or given) a linear demand function of the form $Q=\frac{a}{b}-\frac1bP$, or its equivalent inverse form $P=a-bQ$, where $a$ and $b$ are positive numbers.

Given that price elasticity of demand at the optimum is $-1$, which you were right to point out (assuming linear demand), you can use the elasticity formula
\begin{equation}
-1=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}=-\frac1b\cdot\frac{P}{Q}
\end{equation}
and the demand equation to solve for the two unknowns $P$ and $Q$.