The system of equations
\[\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 5\]
has exactly one solution. What is $z$ in this solution?
Taking reciprocals, we get
\begin{align*}
\frac{1}{x} + \frac{1}{y} &= 1, \\
\frac{1}{x} + \frac{1}{z} &= 1, \\
\frac{1}{y} + \frac{1}{z} &= \frac{1}{5}.
\end{align*}Then $\frac{1}{z} = \frac{1}{x+z}-\frac{1}{x} = \frac{1}{y+z}-\frac{1}{y} = \frac{1}{5},$ so $z = \boxed{5}.$