To find the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-4, 3)\)[/tex], we'll follow these steps:
1. Identify the slope of the given line:
The given line's equation is in point-slope form: [tex]\( y - 3 = -2(x + 4) \)[/tex]. The coefficient of [tex]\( x \)[/tex] in this equation is the slope, which is [tex]\(-2\)[/tex].
2. Calculate the slope of the perpendicular line:
The slope of the line perpendicular to another line is the negative reciprocal of the slope of the given line. If the slope of the given line is [tex]\( m \)[/tex], the slope of the perpendicular line [tex]\( m' \)[/tex] is given by [tex]\( m' = -\frac{1}{m} \)[/tex].
Given slope [tex]\( m = -2 \)[/tex], the perpendicular slope [tex]\( m' = -\frac{1}{-2} = \frac{1}{2} \)[/tex].
3. Use the slope and point to write the equation:
The equation of the line in point-slope form is given by:
[tex]\[
y - y_1 = m'(x - x_1)
\][/tex]
Substituting the point [tex]\((-4, 3)\)[/tex] and the perpendicular slope [tex]\( \frac{1}{2} \)[/tex] into the equation, we get:
[tex]\[
y - 3 = \frac{1}{2}(x + 4)
\][/tex]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-4, 3)\)[/tex] is:
[tex]\[
y - 3 = \frac{1}{2}(x + 4)
\][/tex]
