## Problem Statement

Find the point of intersection of the line and the plane.

$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2} \\
& x-3 y+7 z-24=0
\end{aligned}
$$
## Solution

Let's write the parametric equations of the line.

$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2}=t \Rightarrow \\
& \left\{\begin{array}{l}
x=1-t \\
y=-5+4 t \\
z=1+2 t
\end{array}\right.
\end{aligned}
$$

Substitute into the equation of the plane:

$$
\begin{aligned}
& (1-t)-3(-5+4 t)+7(1+2 t)-24=0 \\
& 1-t+15-12 t+7+14 t-24=0 \\
& t-1=0 \\
& t=1
\end{aligned}
$$

Find the coordinates of the intersection point of the line and the plane:

$$
\left\{\begin{array}{l}
x=1-1=0 \\
y=-5+4 \cdot 1=-1 \\
z=1+2 \cdot 1=3
\end{array}\right.
$$

We get:

$(0 ;-1 ; 3)$

## Problem Kuznetsov Analytic Geometry 14-3