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,...
Related questions
Question
![**Problem Statement:**
Write an equation for a line perpendicular to \( y = -3x - 3 \) and passing through the point \( (6,1) \).
**Solution:**
To find the equation of a line perpendicular to a given line, we need to determine the slope of the perpendicular line.
1. **Identify the slope of the given line:**
The equation of the given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. For the line \( y = -3x - 3 \), the slope \( m = -3 \).
2. **Determine the slope of the perpendicular line:**
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the given line is \( -3 \), the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{-3} = \frac{1}{3}
\]
3. **Use the point-slope form to find the equation:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Using the point \( (6,1) \) and the slope \( \frac{1}{3} \), the equation becomes:
\[
y - 1 = \frac{1}{3}(x - 6)
\]
4. **Convert to slope-intercept form:**
\[
y - 1 = \frac{1}{3}x - 2
\]
\[
y = \frac{1}{3}x - 2 + 1
\]
\[
y = \frac{1}{3}x - 1
\]
**Final Equation:**
The equation of the line perpendicular to \( y = -3x - 3 \) and passing through the point \( (6,1) \) is:
\[
y = \frac{1}{3}x - 1
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe84762dd-baf9-4e17-b4ea-3980bb4862e9%2Fd57cce37-90be-40e9-a074-ac78ad1f3aa1%2F5ajwkg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write an equation for a line perpendicular to \( y = -3x - 3 \) and passing through the point \( (6,1) \).
**Solution:**
To find the equation of a line perpendicular to a given line, we need to determine the slope of the perpendicular line.
1. **Identify the slope of the given line:**
The equation of the given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. For the line \( y = -3x - 3 \), the slope \( m = -3 \).
2. **Determine the slope of the perpendicular line:**
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the given line is \( -3 \), the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{-3} = \frac{1}{3}
\]
3. **Use the point-slope form to find the equation:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Using the point \( (6,1) \) and the slope \( \frac{1}{3} \), the equation becomes:
\[
y - 1 = \frac{1}{3}(x - 6)
\]
4. **Convert to slope-intercept form:**
\[
y - 1 = \frac{1}{3}x - 2
\]
\[
y = \frac{1}{3}x - 2 + 1
\]
\[
y = \frac{1}{3}x - 1
\]
**Final Equation:**
The equation of the line perpendicular to \( y = -3x - 3 \) and passing through the point \( (6,1) \) is:
\[
y = \frac{1}{3}x - 1
\]
Expert Solution
Step 1
The equation of the line passing through the point (x0 , y0 ) with slope m is
y - y0 = m ( x - x0)
Step by step
Solved in 2 steps with 1 images
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