[Coordinate method on the plane]

Find the equation of the line passing through the point $M(-1 ; 4)$ and perpendicular to the line $x-2 y+4=0$.

#
Apply the condition of perpendicularity of two lines ( $\left.k_{1} \cdot k_{2}=-1\right)$.

## Solution

The equation of the given line can be written as $y=\frac{1}{2} x+2$. Then its slope $k_{1}=\frac{1}{2}$. If $k_{2}$ is the slope of the line perpendicular to the given one, then $k_{1} \cdot k_{2}=-1$. Therefore,

$$
k_{2}=-\frac{1}{k_{1}}=-2
$$

We will find the equation of the desired line using the point $M(-1 ; 4)$ and the slope $k_{2}=-2$:

$$
y-4=-2(x+1)
$$

Let's write the equation of this line in general form:

$$
2 x+y-2=0
$$

## Answer

$2 x+y-2=0$.