What is the equation of the line that is perpendicular to and has the same [tex]$y$[/tex]-intercept as the given line?

Given line: [tex]\( y = \frac{1}{5}x + 1 \)[/tex]

A. [tex]\( y = \frac{1}{5}x + 1 \)[/tex]
B. [tex]\( y = \frac{1}{5}x + 5 \)[/tex]
C. [tex]\( y = 5x + 1 \)[/tex]
D. [tex]\( y = 5x + 5 \)[/tex]



Answer :

To determine the equation of the line that is perpendicular to the given line and has the same [tex]\( y \)[/tex]-intercept, we need to follow these steps:

1. Identify the slope of the given line:
The equation of the given line is:
[tex]\[ y = \frac{1}{5}x + 1 \][/tex]
Here, the slope ([tex]\(m\)[/tex]) is [tex]\(\frac{1}{5}\)[/tex].

2. Find the perpendicular slope:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the given line. Therefore, the perpendicular slope is:
[tex]\[ -\frac{1}{\frac{1}{5}} = -5 \][/tex]

3. Identify the [tex]\( y \)[/tex]-intercept of the given line:
The [tex]\( y \)[/tex]-intercept of the given line is 1, as seen in the equation [tex]\(y = \frac{1}{5}x + 1\)[/tex].

4. Formulate the equation of the perpendicular line:
Utilizing the perpendicular slope found in step 2 and the [tex]\( y \)[/tex]-intercept from step 3, the equation of the line perpendicular to the given line, which also has the same [tex]\( y \)[/tex]-intercept, is:
[tex]\[ y = -5x + 1 \][/tex]

5. Compare with the given options:
We need to compare this equation with the options provided to see which matches:

- [tex]\( y = \frac{1}{5}x + 1 \)[/tex] (This has the same [tex]\( y \)[/tex]-intercept but is not perpendicular)
- [tex]\( y = \frac{1}{5}x + 5 \)[/tex] (This has a different [tex]\( y \)[/tex]-intercept and is not perpendicular)
- [tex]\( y = 5x + 1 \)[/tex] (This is not correct; although it has the same [tex]\( y \)[/tex]-intercept, the slope is incorrect)
- [tex]\( y = 5x + 5 \)[/tex] (This also has a different [tex]\( y \)[/tex]-intercept and is not perpendicular)

After considering all the options, none of these match our equation [tex]\(y = -5x + 1\)[/tex].

Hence, there is no equation in the provided options that satisfies the conditions of being perpendicular to the given line [tex]\( y = \frac{1}{5}x + 1 \)[/tex] and having the same [tex]\( y \)[/tex]-intercept. Thus, the correct answer is:

[tex]\[ \boxed{\text{None}} \][/tex]