To find the equation of the line that is perpendicular to a given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line:
The equation of the given line is \( y + 3 = -4(x + 4) \). We can rewrite this in slope-intercept form ( \( y = mx + b \) ) to easily identify the slope \( m \).
Let's do that by isolating \( y \):
[tex]\[
y + 3 = -4(x + 4)
\][/tex]
[tex]\[
y + 3 = -4x - 16
\][/tex]
[tex]\[
y = -4x - 16 - 3
\][/tex]
[tex]\[
y = -4x - 19
\][/tex]
Thus, the slope \( m \) of the given line is \( -4 \).
2. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of \( -4 \) is:
[tex]\[
\frac{1}{-(-4)} = \frac{1}{4}
\][/tex]
So, the slope of the perpendicular line is \( \frac{1}{4} \).
3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line.
We are given the point \( (-4, -3) \) and the slope \( \frac{1}{4} \). Substituting these values into the point-slope form, we get:
[tex]\[
y - (-3) = \frac{1}{4}(x - (-4))
\][/tex]
Simplifying the expression:
[tex]\[
y + 3 = \frac{1}{4}(x + 4)
\][/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]\( y + 3 = \frac{1}{4}(x + 4) \)[/tex].
