Question

Passing through left parenthesis 5 comma negative 4 right parenthesis and perpendicular to the line whose equation is x minus 5 y minus 9 equals 0

the equation of the line is

Answer 1

To find the equation of the line passing through the point \( (5, -4) \) and perpendicular to the line given by the equation \( x - 5y - 9 = 0 \), we can follow these steps:

1. **Find the slope of the given line**:
The equation can be rewritten in slope-intercept form \( y = mx + b \):
\[
x - 5y - 9 = 0 \implies 5y = x - 9 \implies y = \frac{1}{5}x - \frac{9}{5}
\]
From this, we see that the slope \( m \) of the given line is \( \frac{1}{5} \).

2. **Find the slope of the perpendicular line**:
The slope of a line that is perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the slope \( m' \) of the line we want to find is:
\[
m' = -\frac{1}{\frac{1}{5}} = -5
\]

3. **Use the point-slope form to find the equation of the line**:
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (5, -4) \) and \( m = -5 \).
Substituting the values:
\[
y - (-4) = -5(x - 5)
\]
Simplifying this:
\[
y + 4 = -5x + 25
\]
\[
y = -5x + 25 - 4
\]
\[
y = -5x + 21
\]

Thus, the equation of the line passing through the point \( (5, -4) \) and perpendicular to the given line is:
\[
\boxed{y = -5x + 21}
\]