The linear Turán number of the 3-graph $P_5$

We prove that for any linear 3-graph on $n$ vertices without a path of length 5, the number of edges is at most $\frac{15}{11}n$, and the equality holds if and only if the graph is the disjoint union of $G_0$, a graph with 11 vertices and 15 edges. Thus, $e{x}_L(n,P_5)\le \frac{15}{11}n$, and the equality holds if and only if $11\mathrm{\mid }n$.