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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![**Problem Statement:**
Find the equation of the line in slope-intercept form that is parallel to \(3x - 4y = 2\) and goes through the point \((-12, 1)\).
**Solution Explanation:**
1. **Convert the Given Line to Slope-Intercept Form:**
Start by rewriting the equation \(3x - 4y = 2\) in the slope-intercept form \(y = mx + b\), where \(m\) is the slope.
\[
3x - 4y = 2 \implies -4y = -3x + 2 \implies y = \frac{3}{4}x - \frac{1}{2}
\]
The slope \(m\) of the original line is \(\frac{3}{4}\).
2. **Determine the Slope of the Parallel Line:**
Parallel lines have the same slope. Therefore, the slope of the new line will also be \(\frac{3}{4}\).
3. **Use the Point-Slope Formula to Find the Equation:**
The point-slope formula is:
\[
y - y_1 = m(x - x_1)
\]
Plug in the slope \(m = \frac{3}{4}\) and the point \((-12, 1)\).
\[
y - 1 = \frac{3}{4}(x + 12)
\]
4. **Simplify to Slope-Intercept Form:**
\[
y - 1 = \frac{3}{4}x + \frac{36}{4}
\]
\[
y - 1 = \frac{3}{4}x + 9
\]
\[
y = \frac{3}{4}x + 10
\]
**Result:**
The equation of the line parallel to \(3x - 4y = 2\) and passing through the point \((-12, 1)\) is:
\[ y = \frac{3}{4}x + 10 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe84762dd-baf9-4e17-b4ea-3980bb4862e9%2F1c89e229-3557-48ea-810d-f4f631df8103%2F232jv3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the line in slope-intercept form that is parallel to \(3x - 4y = 2\) and goes through the point \((-12, 1)\).
**Solution Explanation:**
1. **Convert the Given Line to Slope-Intercept Form:**
Start by rewriting the equation \(3x - 4y = 2\) in the slope-intercept form \(y = mx + b\), where \(m\) is the slope.
\[
3x - 4y = 2 \implies -4y = -3x + 2 \implies y = \frac{3}{4}x - \frac{1}{2}
\]
The slope \(m\) of the original line is \(\frac{3}{4}\).
2. **Determine the Slope of the Parallel Line:**
Parallel lines have the same slope. Therefore, the slope of the new line will also be \(\frac{3}{4}\).
3. **Use the Point-Slope Formula to Find the Equation:**
The point-slope formula is:
\[
y - y_1 = m(x - x_1)
\]
Plug in the slope \(m = \frac{3}{4}\) and the point \((-12, 1)\).
\[
y - 1 = \frac{3}{4}(x + 12)
\]
4. **Simplify to Slope-Intercept Form:**
\[
y - 1 = \frac{3}{4}x + \frac{36}{4}
\]
\[
y - 1 = \frac{3}{4}x + 9
\]
\[
y = \frac{3}{4}x + 10
\]
**Result:**
The equation of the line parallel to \(3x - 4y = 2\) and passing through the point \((-12, 1)\) is:
\[ y = \frac{3}{4}x + 10 \]
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