To determine the equation of a line that is perpendicular to a given line and has the same [tex]\( y \)[/tex]-intercept, follow these steps:
### Step 1: Identify the slope of the given line
The given lines are:
1. [tex]\( y = \frac{1}{5}x + 1 \)[/tex]
2. [tex]\( y = \frac{1}{5}x + 5 \)[/tex]
3. [tex]\( y = 5x + 1 \)[/tex]
4. [tex]\( y = 5x + 5 \)[/tex]
We need to work with the first equation, [tex]\( y = \frac{1}{5}x + 1 \)[/tex], to find the slope and [tex]\( y \)[/tex]-intercept of the perpendicular line.
### Step 2: Determine the slope of the perpendicular line
The slope of the given line [tex]\( y = \frac{1}{5}x + 1 \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. The negative reciprocal of [tex]\( \frac{1}{5} \)[/tex] is [tex]\( -5 \)[/tex].
### Step 3: Determine the [tex]\( y \)[/tex]-intercept of the perpendicular line
The [tex]\( y \)[/tex]-intercept of our given line is [tex]\( 1 \)[/tex]. Since we want the perpendicular line to have the same [tex]\( y \)[/tex]-intercept, we use [tex]\( 1 \)[/tex] as the [tex]\( y \)[/tex]-intercept for the perpendicular line as well.
### Step 4: Write the equation of the perpendicular line
We now have the slope [tex]\( -5 \)[/tex] and the [tex]\( y \)[/tex]-intercept [tex]\( 1 \)[/tex]. Plugging these values into the slope-intercept form [tex]\( y = mx + b \)[/tex], we get:
[tex]\[ y = -5x + 1 \][/tex]
### Conclusion
The equation of the line that is perpendicular to [tex]\( y = \frac{1}{5} x + 1 \)[/tex] and has the same [tex]\( y \)[/tex]-intercept is:
[tex]\[ y = -5x + 1 \][/tex]
Thus, the correct answer is:
[tex]\[ y = -5x + 1 \][/tex]
However, it looks like this form doesn't directly appear in the original options.
Since it is required to choose the corresponding correct option from given options:
The correct line that is perpendicular to the given one and has the correct [tex]\( y \)[/tex]-intercept is indeed not among the provided choices, possibly it needs to be verified as the intended correct perpendicular line equation here is:
[tex]\[ y = -5x + 1 \][/tex]
