Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Solve the following set of equations by graphing:**
\[ y + x = 9 \]
\[ 3y = 2x + 8 \]
**Options:**
- (3.75, 5.25)
- (3.5, 5.5)
- No solutions
- (3.8, 5.2)
To solve these equations by graphing, we first need to rearrange both equations into slope-intercept form (\( y = mx + b \)).
1. For the first equation \( y + x = 9 \), we can rearrange it as follows:
\[ y = 9 - x \]
2. For the second equation \( 3y = 2x + 8 \), we can rearrange it as follows:
\[ y = \frac{2}{3}x + \frac{8}{3} \]
### Explanation:
#### Converting the Equations:
- **First Equation:**
\[ y = 9 - x \]
- **Second Equation:**
\[ y = \frac{2}{3}x + \frac{8}{3} \]
#### Steps to Graph:
1. **Graph the first line:**
- The y-intercept (\(b\)) is 9.
- The slope (\(m\)) is -1, meaning for every 1 unit increase in \(x\), \(y\) decreases by 1.
2. **Graph the second line:**
- The y-intercept is \(\frac{8}{3}\) (approximately 2.67).
- The slope is \(\frac{2}{3}\), meaning for every 3 unit increase in \(x\), \(y\) increases by 2.
**Finding the Intersecting Point:**
- To find the intersection point, you can solve the equations simultaneously by setting them equal:
\[ 9 - x = \frac{2}{3}x + \frac{8}{3} \]
- Solve for \(x\):
\[ 27 - 3x = 2x + 8 \]
\[ 27 - 8 = 5x \]
\[ 19 = 5x \]
\[ x = \frac{19}{5} \approx 3.8 \]
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f130582-8262-45ba-b1cd-6680cc0034f3%2F7853756a-746e-47de-a8a4-5cb7cf794de2%2Flpy8v9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve the following set of equations by graphing:**
\[ y + x = 9 \]
\[ 3y = 2x + 8 \]
**Options:**
- (3.75, 5.25)
- (3.5, 5.5)
- No solutions
- (3.8, 5.2)
To solve these equations by graphing, we first need to rearrange both equations into slope-intercept form (\( y = mx + b \)).
1. For the first equation \( y + x = 9 \), we can rearrange it as follows:
\[ y = 9 - x \]
2. For the second equation \( 3y = 2x + 8 \), we can rearrange it as follows:
\[ y = \frac{2}{3}x + \frac{8}{3} \]
### Explanation:
#### Converting the Equations:
- **First Equation:**
\[ y = 9 - x \]
- **Second Equation:**
\[ y = \frac{2}{3}x + \frac{8}{3} \]
#### Steps to Graph:
1. **Graph the first line:**
- The y-intercept (\(b\)) is 9.
- The slope (\(m\)) is -1, meaning for every 1 unit increase in \(x\), \(y\) decreases by 1.
2. **Graph the second line:**
- The y-intercept is \(\frac{8}{3}\) (approximately 2.67).
- The slope is \(\frac{2}{3}\), meaning for every 3 unit increase in \(x\), \(y\) increases by 2.
**Finding the Intersecting Point:**
- To find the intersection point, you can solve the equations simultaneously by setting them equal:
\[ 9 - x = \frac{2}{3}x + \frac{8}{3} \]
- Solve for \(x\):
\[ 27 - 3x = 2x + 8 \]
\[ 27 - 8 = 5x \]
\[ 19 = 5x \]
\[ x = \frac{19}{5} \approx 3.8 \]
-
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