Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Understanding Linear Equations: Parallel and Intersecting Lines
In this example, we are given two linear equations:
\[ y = \frac{3}{5}x + 7 \]
and
\[ 3y - 5x = 3 \]
#### Analyzing the Equations
1. **First Equation**:
\[ y = \frac{3}{5}x + 7 \]
This equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m\) is \(\frac{3}{5}\) and the y-intercept \(b\) is 7.
2. **Second Equation**:
\[ 3y - 5x = 3 \]
To analyze this equation further, we convert it into the slope-intercept form.
\[ 3y = 5x + 3 \]
Dividing through by 3 to isolate \(y\):
\[ y = \frac{5}{3}x + 1 \]
After conversion, the slope \(m\) is \(\frac{5}{3}\) and the y-intercept \(b\) is 1.
#### Comparison of Slopes
For both equations, we observe:
- **First Equation's Slope**: \(\frac{3}{5}\)
- **Second Equation's Slope (after rearranging)**: \(\frac{5}{3}\)
Since \(\frac{3}{5}\) is not equal to \(\frac{5}{3}\), these lines are not parallel and will intersect at a point.
#### Conclusion
The lines represented by the equations \( y = \frac{3}{5}x + 7 \) and \( 3y - 5x = 3 \) are **intersecting** lines because their slopes (\(\frac{3}{5}\) and \(\frac{5}{3}\)) are not equal. They will meet at one specific point in the coordinate plane.
### Visual Representation
_(Optional if using the website: A graph can be included to visually show where the lines intersect. Each line can be plotted using its slope and y-intercept to demonstrate their intersection visibly.)_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e24649-3abe-463a-ac67-26fbe087aa75%2Fd6de3746-e932-48a8-80f4-075eb5caed15%2F7tkaabl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Linear Equations: Parallel and Intersecting Lines
In this example, we are given two linear equations:
\[ y = \frac{3}{5}x + 7 \]
and
\[ 3y - 5x = 3 \]
#### Analyzing the Equations
1. **First Equation**:
\[ y = \frac{3}{5}x + 7 \]
This equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m\) is \(\frac{3}{5}\) and the y-intercept \(b\) is 7.
2. **Second Equation**:
\[ 3y - 5x = 3 \]
To analyze this equation further, we convert it into the slope-intercept form.
\[ 3y = 5x + 3 \]
Dividing through by 3 to isolate \(y\):
\[ y = \frac{5}{3}x + 1 \]
After conversion, the slope \(m\) is \(\frac{5}{3}\) and the y-intercept \(b\) is 1.
#### Comparison of Slopes
For both equations, we observe:
- **First Equation's Slope**: \(\frac{3}{5}\)
- **Second Equation's Slope (after rearranging)**: \(\frac{5}{3}\)
Since \(\frac{3}{5}\) is not equal to \(\frac{5}{3}\), these lines are not parallel and will intersect at a point.
#### Conclusion
The lines represented by the equations \( y = \frac{3}{5}x + 7 \) and \( 3y - 5x = 3 \) are **intersecting** lines because their slopes (\(\frac{3}{5}\) and \(\frac{5}{3}\)) are not equal. They will meet at one specific point in the coordinate plane.
### Visual Representation
_(Optional if using the website: A graph can be included to visually show where the lines intersect. Each line can be plotted using its slope and y-intercept to demonstrate their intersection visibly.)_
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