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...
Related questions
Question
![### Solving a System of Equations by Substitution
#### Problem Statement
Solve the following system of equations by substitution:
\[0.5r + 0.75s = 10\]
\[r - s = 4\]
#### Given Options
- (6, 2)
- Ø (the null set)
- \(\left(\frac{10\frac{2}{5}}, \frac{-6\frac{2}{5}}\right)\)
- \(\left(\frac{9\frac{4}{5}}, \frac{-6\frac{4}{5}}\right)\)
### Steps to Solve
We start by expressing one variable in terms of the other using the second equation because it is easier to manipulate:
\[ r = s + 4 \]
Next, substitute \( r \) in the first equation:
\[ 0.5(s + 4) + 0.75s = 10 \]
Then, solve this equation for \( s \):
\[ 0.5s + 2 + 0.75s = 10 \]
\[ 1.25s + 2 = 10 \]
\[ 1.25s = 8 \]
\[ s = \frac{8}{1.25} = 6.4 \]
Now, substitute \( s = 6.4 \) back into the expression \( r = s + 4 \):
\[ r = 6.4 + 4 = 10.4 \]
So, our solution to the system of equations is:
\[ (r, s) = (10.4, 6.4) \]
Examining the given options, none of them display the exact value of (10.4, 6.4). Therefore, we need to select the closest representation or analyze further if this problem is meant to show that none of the given solutions match.
### Graphical Explanation (if applicable)
The graphical explanation would involve plotting the lines represented by the equations \(0.5r + 0.75s = 10\) and \(r - s = 4\). The point where these two lines intersect represents the solution to the system of equations.
Upon inspection of the provided answer choices:
- (6, 2)
- Ø (the null set)
- \(\left(\frac{10\frac{2}{5}}, \frac{-6\frac{2}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb4c9257-c3a4-42ab-bf0e-0de9c4c5a001%2Feda07e54-e850-4ac6-8d6e-bf7c06f233d7%2F8qmklfe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a System of Equations by Substitution
#### Problem Statement
Solve the following system of equations by substitution:
\[0.5r + 0.75s = 10\]
\[r - s = 4\]
#### Given Options
- (6, 2)
- Ø (the null set)
- \(\left(\frac{10\frac{2}{5}}, \frac{-6\frac{2}{5}}\right)\)
- \(\left(\frac{9\frac{4}{5}}, \frac{-6\frac{4}{5}}\right)\)
### Steps to Solve
We start by expressing one variable in terms of the other using the second equation because it is easier to manipulate:
\[ r = s + 4 \]
Next, substitute \( r \) in the first equation:
\[ 0.5(s + 4) + 0.75s = 10 \]
Then, solve this equation for \( s \):
\[ 0.5s + 2 + 0.75s = 10 \]
\[ 1.25s + 2 = 10 \]
\[ 1.25s = 8 \]
\[ s = \frac{8}{1.25} = 6.4 \]
Now, substitute \( s = 6.4 \) back into the expression \( r = s + 4 \):
\[ r = 6.4 + 4 = 10.4 \]
So, our solution to the system of equations is:
\[ (r, s) = (10.4, 6.4) \]
Examining the given options, none of them display the exact value of (10.4, 6.4). Therefore, we need to select the closest representation or analyze further if this problem is meant to show that none of the given solutions match.
### Graphical Explanation (if applicable)
The graphical explanation would involve plotting the lines represented by the equations \(0.5r + 0.75s = 10\) and \(r - s = 4\). The point where these two lines intersect represents the solution to the system of equations.
Upon inspection of the provided answer choices:
- (6, 2)
- Ø (the null set)
- \(\left(\frac{10\frac{2}{5}}, \frac{-6\frac{2}{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning