To find the equation of the line that is perpendicular to the given line [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 given line is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For the line [tex]\( y = \frac{4}{5}x + 23 \)[/tex], the slope [tex]\( m \)[/tex] 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. Therefore, the slope of the perpendicular line is [tex]\( -\frac{1}{\frac{4}{5}} \)[/tex], which simplifies to [tex]\( -\frac{5}{4} \)[/tex].
3. Use the point-slope form to find the equation:
The point-slope form of a line's equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. We have the point [tex]\((-40, 20)\)[/tex] and the slope [tex]\( -\frac{5}{4} \)[/tex].
Plugging in these values, we get:
[tex]\[
y - 20 = -\frac{5}{4}(x + 40)
\][/tex]
4. Simplify the equation:
Distribute the slope on the right-hand side:
[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]
[tex]\[
y = -\frac{5}{4}x - 30
\][/tex]
So, the equation of the line that is perpendicular to [tex]\( y = \frac{4}{5}x + 23 \)[/tex] and passes through the point [tex]\((-40, 20)\)[/tex] is:
[tex]\[
y = -\frac{5}{4}x - 30
\][/tex]
Therefore, the correct answer is:
B. [tex]\( y = -\frac{5}{4}x - 30 \)[/tex]
