Find the point where $\displaystyle \frac{x-2}{5} = y = \frac{z+1}{3} $ intersects $\displaystyle x+y-z=2. $
Write the line as $\displaystyle \left\{ \begin{gathered} x(t) = 5t + 2 \hfill \\
y(t) = t \hfill \\ z(t) = 3t - 1 \hfill \\ \end{gathered} \right.$
Now substitute into the plane and then solve for t.