Sure, let's work through the problem step-by-step to find the equation of the line that is perpendicular and has the same [tex]\( y \)[/tex]-intercept as the given line.
### Step 1: Understanding the equations
We are given four equations:
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]
### Step 2: Identifying the [tex]\( y \)[/tex]-intercepts
For an equation in the form [tex]\( y = mx + b \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\( b \)[/tex].
From the given equations:
- [tex]\( y = \frac{1}{5}x + 1 \)[/tex]: [tex]\( y \)[/tex]-intercept is [tex]\( 1 \)[/tex]
- [tex]\( y = \frac{1}{5}x + 5 \)[/tex]: [tex]\( y \)[/tex]-intercept is [tex]\( 5 \)[/tex]
- [tex]\( y = 5x + 1 \)[/tex]: [tex]\( y \)[/tex]-intercept is [tex]\( 1 \)[/tex]
- [tex]\( y = 5x + 5 \)[/tex]: [tex]\( y \)[/tex]-intercept is [tex]\( 5 \)[/tex]
So, the [tex]\( y \)[/tex]-intercepts are [tex]\( 1 \)[/tex] and [tex]\( 5 \)[/tex].
### Step 3: Finding the slope of a perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
For the equations [tex]\( y = \frac{1}{5}x + 1 \)[/tex] and [tex]\( y = \frac{1}{5}x + 5 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
- The perpendicular slope would be the negative reciprocal of [tex]\( \frac{1}{5} \)[/tex], which is [tex]\( -5 \)[/tex].
### Step 4: Forming the equations with the [tex]\( y \)[/tex]-intercepts
Now, we need to form equations with slopes of [tex]\( -5 \)[/tex] and the [tex]\( y \)[/tex]-intercepts we identified [tex]\( (1 \text{ and } 5) \)[/tex].
#### For [tex]\( y \)[/tex]-intercept [tex]\( 1 \)[/tex]:
[tex]\[ y = -5x + 1 \][/tex]
#### For [tex]\( y \)[/tex]-intercept [tex]\( 5 \)[/tex]:
[tex]\[ y = -5x + 5 \][/tex]
### Step 5: Matching the final equation
Comparing these formed equations to the given ones:
- [tex]\( y = 5x + 1 \)[/tex] has the same [tex]\( y \)[/tex]-intercept [tex]\( 1 \)[/tex] but does not match the slope [tex]\( -5 \)[/tex] requirement.
- The other equations also do not match both the perpendicular slope and given [tex]\( y \)[/tex]-intercepts.
Therefore, the equation that matches being perpendicular and having the same [tex]\( y \)[/tex]-intercept as one of the given lines is:
The equation of the line [tex]\( y = 5x + 1 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
