## Problem Statement

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

$\frac{x-5}{-2}=\frac{y-2}{0}=\frac{z+4}{-1}$

$2 x-5 y+4 z+24=0$
## Solution

Let's write the parametric equations of the line.

$$
\begin{aligned}
& \frac{x-5}{-2}=\frac{y-2}{0}=\frac{z+4}{-1}=t \Rightarrow \\
& \left\{\begin{array}{l}
x=5-2 t \\
y=2 \\
z=-4-t
\end{array}\right.
\end{aligned}
$$

Substitute into the equation of the plane:

$2(5-2 t)-5 \cdot 2+4(-4-t)+24=0$

$10-4 t-10-16-4 t+24=0$

$-8 t+8=0$

$t=1$

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

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

We get:

$(3 ; 2 ;-5)$

## Problem Kuznetsov Analytic Geometry 14-19