  This article provides bounds on the size of a 3-uniform linear hypergraph with restricted matching number and maximum degree. In particular, we show that if a 3-uniform, linear family $\mathcal{F}$ has maximum matching size $\nu$ and maximum degree $\Delta$ such that $\Delta\geq \frac{23}{6}\nu(1+\frac{1}{\nu-1})$, then $|\mathcal{F}|\leq \Delta \nu$. 
Size of a 3-uniform linear hypergraph