To determine which equation represents the line that is perpendicular to [tex]\( y = \frac{4}{5}x + 23 \)[/tex] and passes through the point [tex]\((-40, 20)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The equation of the given line is [tex]\( y = \frac{4}{5}x + 23 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\(\frac{4}{5}\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. For the given slope [tex]\(\frac{4}{5}\)[/tex], the perpendicular slope [tex]\( m_{\perpendicular} \)[/tex] is:
[tex]\[
m_{\perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{4}{5}} = -\frac{5}{4}
\][/tex]
3. Use the point-slope form to find the equation of the line:
The point-slope form of the equation of a line is [tex]\( y - y_1 = m (x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes and [tex]\( m \)[/tex] is the slope.
Given point [tex]\((-40, 20)\)[/tex] and slope [tex]\(-\frac{5}{4}\)[/tex], the equation becomes:
[tex]\[
y - 20 = -\frac{5}{4} (x + 40)
\][/tex]
4. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 20 = -\frac{5}{4} (x + 40)
\][/tex]
Distribute the slope [tex]\(-\frac{5}{4}\)[/tex]:
[tex]\[
y - 20 = -\frac{5}{4}x - 50
\][/tex]
Add 20 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{5}{4}x - 50 + 20
\][/tex]
Simplify the constants:
[tex]\[
y = -\frac{5}{4}x - 30
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = \frac{4}{5}x + 23 \)[/tex] and passing through [tex]\((-40, 20)\)[/tex] is [tex]\( y = -\frac{5}{4}x - 30 \)[/tex].
So, the correct answer is:
B. [tex]\( y = -\frac{5}{4}x - 30 \)[/tex]
