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
100%
How do I do this?
![### Solving a System of Equations Using Augmented Matrix Methods
#### Problem Statement:
Solve the system of equations using augmented matrix methods.
\[
x_1 - 2x_2 = -1
\]
\[
2x_1 - x_2 = 7
\]
#### Explanation:
To solve this system using the augmented matrix method, follow these steps:
1. **Write the augmented matrix**: Convert the system of equations into an augmented matrix.
\[
\begin{bmatrix}
1 & -2 & | & -1 \\
2 & -1 & | & 7
\end{bmatrix}
\]
2. **Row operations**: Use row operations to transform the matrix into row-echelon form (or reduced row-echelon form).
- First, make the first element in the first column a leading 1 (which it already is), and use it to make all elements below it zero.
- Subtract 2 times the first row from the second row.
\[
\begin{bmatrix}
1 & -2 & | & -1 \\
0 & 3 & | & 9
\end{bmatrix}
\]
3. **Back substitution**: Once the matrix is in row-echelon form, solve for the variables starting from the last row.
- From the second row \(0x_1 + 3x_2 = 9\), solve for \(x_2\):
\[
x_2 = \frac{9}{3} = 3
\]
- Substitute \(x_2 = 3\) into the first equation \(x_1 - 2x_2 = -1\) to solve for \(x_1\):
\[
x_1 - 2(3) = -1 \\
x_1 - 6 = -1 \\
x_1 = 5
\]
#### Solution:
\(x_1 = 5\) and \(x_2 = 3\).
This concludes the solution using the augmented matrix method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7905efcd-5a11-4079-98fb-e5b3e7ed71d0%2F24007519-b3d0-4455-91ed-12a40ccb3371%2Fosaa07_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a System of Equations Using Augmented Matrix Methods
#### Problem Statement:
Solve the system of equations using augmented matrix methods.
\[
x_1 - 2x_2 = -1
\]
\[
2x_1 - x_2 = 7
\]
#### Explanation:
To solve this system using the augmented matrix method, follow these steps:
1. **Write the augmented matrix**: Convert the system of equations into an augmented matrix.
\[
\begin{bmatrix}
1 & -2 & | & -1 \\
2 & -1 & | & 7
\end{bmatrix}
\]
2. **Row operations**: Use row operations to transform the matrix into row-echelon form (or reduced row-echelon form).
- First, make the first element in the first column a leading 1 (which it already is), and use it to make all elements below it zero.
- Subtract 2 times the first row from the second row.
\[
\begin{bmatrix}
1 & -2 & | & -1 \\
0 & 3 & | & 9
\end{bmatrix}
\]
3. **Back substitution**: Once the matrix is in row-echelon form, solve for the variables starting from the last row.
- From the second row \(0x_1 + 3x_2 = 9\), solve for \(x_2\):
\[
x_2 = \frac{9}{3} = 3
\]
- Substitute \(x_2 = 3\) into the first equation \(x_1 - 2x_2 = -1\) to solve for \(x_1\):
\[
x_1 - 2(3) = -1 \\
x_1 - 6 = -1 \\
x_1 = 5
\]
#### Solution:
\(x_1 = 5\) and \(x_2 = 3\).
This concludes the solution using the augmented matrix method.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 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