Answer :
To determine which of the given equations represents a line that is perpendicular to the equation \( y = -6x + 7 \), we must first understand how the slopes of perpendicular lines are related.
### Step-by-Step Solution
#### Step 1: Identify the slope of the given line
The equation of the given line is:
[tex]\[ y = -6x + 7 \][/tex]
In this equation, the coefficient of \( x \) is the slope. Thus, the slope (m) of this line is:
[tex]\[ m = -6 \][/tex]
#### Step 2: Determine the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if one line has a slope \( m \), the slope of the line perpendicular to it will be:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{m} \][/tex]
Given the slope of the original line is \( m = -6 \):
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-6} = \frac{1}{6} \][/tex]
#### Step 3: Identify the equation with the perpendicular slope
Now, we need to find which among the given options has a slope of \( \frac{1}{6} \):
- Option A: \( y = -6x + 3 \)
- Slope: \( -6 \)
- Option B: \( y = \frac{1}{6}x + 3 \)
- Slope: \( \frac{1}{6} \)
- Option C: \( y = -\frac{1}{6}x + 3 \)
- Slope: \( -\frac{1}{6} \)
- Option D: \( y = 6x + 3 \)
- Slope: \( 6 \)
#### Step 4: Choose the correct option
From the above analysis, we can see that the equation with the slope \( \frac{1}{6} \) is:
[tex]\[ \boxed{y = \frac{1}{6}x + 3} \][/tex]
Thus, the equation of the line that is perpendicular to the given line \( y = -6x + 7 \) is:
[tex]\[ \boxed{B. \; y = \frac{1}{6}x + 3} \][/tex]
### Step-by-Step Solution
#### Step 1: Identify the slope of the given line
The equation of the given line is:
[tex]\[ y = -6x + 7 \][/tex]
In this equation, the coefficient of \( x \) is the slope. Thus, the slope (m) of this line is:
[tex]\[ m = -6 \][/tex]
#### Step 2: Determine the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if one line has a slope \( m \), the slope of the line perpendicular to it will be:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{m} \][/tex]
Given the slope of the original line is \( m = -6 \):
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-6} = \frac{1}{6} \][/tex]
#### Step 3: Identify the equation with the perpendicular slope
Now, we need to find which among the given options has a slope of \( \frac{1}{6} \):
- Option A: \( y = -6x + 3 \)
- Slope: \( -6 \)
- Option B: \( y = \frac{1}{6}x + 3 \)
- Slope: \( \frac{1}{6} \)
- Option C: \( y = -\frac{1}{6}x + 3 \)
- Slope: \( -\frac{1}{6} \)
- Option D: \( y = 6x + 3 \)
- Slope: \( 6 \)
#### Step 4: Choose the correct option
From the above analysis, we can see that the equation with the slope \( \frac{1}{6} \) is:
[tex]\[ \boxed{y = \frac{1}{6}x + 3} \][/tex]
Thus, the equation of the line that is perpendicular to the given line \( y = -6x + 7 \) is:
[tex]\[ \boxed{B. \; y = \frac{1}{6}x + 3} \][/tex]
