To determine the equation of a line that is perpendicular to the line [tex]\( y = \frac{4}{5} x + 23 \)[/tex] and passes through the point [tex]\((-40, 20)\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = \frac{4}{5} x + 23 \)[/tex]. The slope ([tex]\( m_1 \)[/tex]) of this line is [tex]\(\frac{4}{5}\)[/tex].
2. Determine the slope of the perpendicular line:
Lines that are perpendicular have slopes that are negative reciprocals of each other. This means if one line has a slope of [tex]\( m \)[/tex], the perpendicular line will have a slope of [tex]\(-\frac{1}{m} \)[/tex].
Therefore, the slope ([tex]\( m_2 \)[/tex]) of the line perpendicular to our given line will be:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{4}{5}} = -\frac{5}{4}
\][/tex]
3. Use the point-slope form to write 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 a point on the line and [tex]\( m \)[/tex] is the slope.
Substituting the point [tex]\((-40, 20)\)[/tex] and the slope [tex]\(-\frac{5}{4}\)[/tex]:
[tex]\[
y - 20 = -\frac{5}{4}(x + 40)
\][/tex]
4. Simplify the equation:
Distribute the slope on the right-hand side:
[tex]\[
y - 20 = -\frac{5}{4}x - \frac{5}{4} \cdot 40
\][/tex]
Simplify inside the parentheses:
[tex]\[
y - 20 = -\frac{5}{4}x - 50
\][/tex]
Finally, solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{5}{4}x - 50 + 20
\][/tex]
[tex]\[
y = -\frac{5}{4}x - 30
\][/tex]
Thus, the equation of the line that is perpendicular to the given line and passes through [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]
