Answer
The point of intersection: $\left( { - 10,1,28} \right)$
Work Step by Step
The line intersects the plane if there exists a parameter value $t$ such that ${\bf{r}}\left( t \right) = \left( {3t + 2,1, - 7t} \right)$ satisfies the equation $2x-3y+z=5$.
We have $x=3t+2$, $y=1$, $z=-7t$. Substituting them in the equation gives
$2x-3y+z=5$
$2(3t+2)-3-7t=5$
$-t+1=5$, ${\ \ \ }$ $t=-4$.
The point corresponding to $t=-4$ is ${\bf{r}}\left( { - 4} \right) = \left( { - 10,1,28} \right)$. Thus, the point $\left( { - 10,1,28} \right)$ is the point of intersection.
