Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Finding the Equation of a Perpendicular Line**
Given the problem statement and provided example:
"Write an equation for a line perpendicular to \( y = 4x + 2 \) and passing through the point \( (8, -6) \)."
### Step-by-Step Solution:
1. **Understand Slopes of Perpendicular Lines:**
- The slope of the given line \( y = 4x + 2 \) is 4.
- For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if \( m_1 = 4 \), then \( m_2 = -\frac{1}{4} \) because \( 4 \times -\frac{1}{4} = -1 \).
2. **Determine the Perpendicular Line's Equation:**
- The slope (\( m \)) of the perpendicular line is \( -\frac{1}{4} \).
- Use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (8, -6) \) and \( m = -\frac{1}{4} \).
Substituting these values, we get:
\[
y - (-6) = -\frac{1}{4}(x - 8)
\]
3. **Simplify the Equation:**
- Rewrite the equation:
\[
y + 6 = -\frac{1}{4}x + 2
\]
- Solve for \( y \):
\[
y = -\frac{1}{4}x + 2 - 6
\]
\[
y = -\frac{1}{4}x - 4
\]
Thus, the equation of the line that is perpendicular to \( y = 4x + 2 \) and passes through the point \( (8, -6) \) is:
\[
y = -\frac{1}{4}x - 4
\]
In the box provided, the final equation should be written as:
\[ y = \]
\[ -\frac{1}{4}x - 4 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F661e79ee-05f0-4794-8531-9b5261a2e126%2F2a95e2bd-4bf5-4b27-875c-efec5cdcd5d1%2Fahiexfk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Equation of a Perpendicular Line**
Given the problem statement and provided example:
"Write an equation for a line perpendicular to \( y = 4x + 2 \) and passing through the point \( (8, -6) \)."
### Step-by-Step Solution:
1. **Understand Slopes of Perpendicular Lines:**
- The slope of the given line \( y = 4x + 2 \) is 4.
- For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if \( m_1 = 4 \), then \( m_2 = -\frac{1}{4} \) because \( 4 \times -\frac{1}{4} = -1 \).
2. **Determine the Perpendicular Line's Equation:**
- The slope (\( m \)) of the perpendicular line is \( -\frac{1}{4} \).
- Use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (8, -6) \) and \( m = -\frac{1}{4} \).
Substituting these values, we get:
\[
y - (-6) = -\frac{1}{4}(x - 8)
\]
3. **Simplify the Equation:**
- Rewrite the equation:
\[
y + 6 = -\frac{1}{4}x + 2
\]
- Solve for \( y \):
\[
y = -\frac{1}{4}x + 2 - 6
\]
\[
y = -\frac{1}{4}x - 4
\]
Thus, the equation of the line that is perpendicular to \( y = 4x + 2 \) and passes through the point \( (8, -6) \) is:
\[
y = -\frac{1}{4}x - 4
\]
In the box provided, the final equation should be written as:
\[ y = \]
\[ -\frac{1}{4}x - 4 \]
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