4.12. Find the point of intersection of the line

$$
(x-2) / 1=(y-3) / 1=(z+1) / 4
$$

and the plane

$$
x+2 y+3 z-10=0
$$
Solution. Let's write down the parametric equations of the given line (formula (4.14)):

$$
x=2+t, y=3+t, z=-1+4 t
$$

Substitute the coordinates of the points on the line into the equation of the plane:

$$
(2+t)+2(3+t)+3(-1+4 t)-10=0, t=1 / 3
$$

The desired coordinates of the intersection point are obtained by substituting the found $t=1 / 3$ into the parametric equations of the line

$$
x_{0}=2+t=7 / 3, y_{0}=3+t=10 / 3, z_{0}=-1+4 t=1 / 3
$$

The intersection point has coordinates $P(7 / 3,10 / 3,1 / 3)$.