jmc

If we multiply the second equation by $-2$, the given equations become $$\begin{array}{rl}3n+m& =14\text{, and}\\ -2n-2m& =-2.\end{array}$$Summing these equations gives $n-m=\boxed{12}$. (Notice that we didn't have to find $n$ or $m$ in order to find $n-m$.)