Question 1: A teacher told her class to write an equation of a line. Jack wrote down the equation 4x + 6y = 8 and Mary wrote down the equation 3x - 7y= 10. The students begin to discuss which line is steeper. Which line is steeper and why?
Question 1: A teacher told her class to write an equation of a line. Jack wrote down the equation 4x + 6y = 8 and Mary wrote down the equation 3x - 7y= 10. The students begin to discuss which line is steeper. Which line is steeper and why?
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 1:**
A teacher told her class to write an equation of a line.
Jack wrote down the equation \(4x + 6y = 8\) and Mary wrote down the equation \(3x - 7y = 10\).
The students begin to discuss which line is steeper. Which line is steeper and why?
---
To determine which line is steeper, we need to compare the slopes of the two lines. To do this, we can rewrite each equation in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope.
1. **Equation 1: \(4x + 6y = 8\)**
- Solve for \(y\):
\[6y = -4x + 8\]
\[y = -\frac{4}{6}x + \frac{8}{6}\]
\[y = -\frac{2}{3}x + \frac{4}{3}\]
- The slope (\(m\)) is \(-\frac{2}{3}\).
2. **Equation 2: \(3x - 7y = 10\)**
- Solve for \(y\):
\[-7y = -3x + 10\]
\[y = \frac{3}{7}x - \frac{10}{7}\]
- The slope (\(m\)) is \(\frac{3}{7}\).
**Comparison:**
- The slope of the first line is \(-\frac{2}{3}\), and the slope of the second line is \(\frac{3}{7}\).
- The absolute value of the first slope is \(\frac{2}{3} \approx 0.667\).
- The absolute value of the second slope is \(\frac{3}{7} \approx 0.429\).
Since \(\frac{2}{3} > \frac{3}{7}\), the line with the equation \(4x + 6y = 8\) is steeper. This is because the absolute value of its slope is greater.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94907f68-16fb-4799-9071-f336e00ec626%2F0064123b-981c-4bc0-97ab-dd802724e205%2F8kmow8e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 1:**
A teacher told her class to write an equation of a line.
Jack wrote down the equation \(4x + 6y = 8\) and Mary wrote down the equation \(3x - 7y = 10\).
The students begin to discuss which line is steeper. Which line is steeper and why?
---
To determine which line is steeper, we need to compare the slopes of the two lines. To do this, we can rewrite each equation in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope.
1. **Equation 1: \(4x + 6y = 8\)**
- Solve for \(y\):
\[6y = -4x + 8\]
\[y = -\frac{4}{6}x + \frac{8}{6}\]
\[y = -\frac{2}{3}x + \frac{4}{3}\]
- The slope (\(m\)) is \(-\frac{2}{3}\).
2. **Equation 2: \(3x - 7y = 10\)**
- Solve for \(y\):
\[-7y = -3x + 10\]
\[y = \frac{3}{7}x - \frac{10}{7}\]
- The slope (\(m\)) is \(\frac{3}{7}\).
**Comparison:**
- The slope of the first line is \(-\frac{2}{3}\), and the slope of the second line is \(\frac{3}{7}\).
- The absolute value of the first slope is \(\frac{2}{3} \approx 0.667\).
- The absolute value of the second slope is \(\frac{3}{7} \approx 0.429\).
Since \(\frac{2}{3} > \frac{3}{7}\), the line with the equation \(4x + 6y = 8\) is steeper. This is because the absolute value of its slope is greater.
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