解:解方程组$\begin {cases}2x + 3y = 3m - 1\\x - y = m - 3\end {cases}$得$\begin {cases}{x = \dfrac {6m - 10}{5}}\\{y = \dfrac {m + 5}{5}}\end {cases}$
根据题意$x < 2,$$y > 1,$列不等式组:
∴$\begin {cases}\dfrac {6m - 10}{5} < 2\\\dfrac {m + 5}{5} > 1\end {cases}$
∴$m $的取值范围是$0 < m < \frac {10}{3}$