jmc

Let $x$ be the smaller integer and $y$ be the larger integer. We have $x+y=30$ and $2y-3x=5$. Solving for $x$ in terms of $y$ using the first equation, we get $x=30-y$. Now we substitute $x$ in terms of $y$ into the second equation. $$\begin{array}{rl}2y-3(30-y)& =5\Rightarrow \\ 2y-90+3y& =5\Rightarrow \\ 5y& =95\Rightarrow \\ y& =19\end{array}$$ And $x=30-19=11$. The positive difference is $y-x=\boxed{8}$.