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Chapter7: Systems Of Equations And Inequalities
Section7.6: Solving Systems With Gaussian Elimination
Problem 1SE: Can any system of linear equations be written as an augmented matrix? Explain why or why not....
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Question
![**Exercises 5–6: Solving Systems of Linear Equations from Augmented Matrices**
In each part of Exercises 5–6, find a system of linear equations in the unknowns \( x_1, x_2, x_3, \ldots \) that corresponds to the given augmented matrix.
Make sure to express each equation clearly and identify the coefficients for the respective variables as depicted by the matrix.
**Example:**
Given the augmented matrix
\[
\begin{bmatrix}
1 & 2 & 3 & | & 4 \\
0 & 5 & 6 & | & 7
\end{bmatrix}
\]
The corresponding system of linear equations is:
\[
\begin{cases}
x_1 + 2x_2 + 3x_3 = 4 \\
5x_2 + 6x_3 = 7
\end{cases}
\]
Continue this process for each provided matrix, deciphering and writing out the respective equations. This practice will enhance your understanding of converting between matrix representations and standard linear equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2Ff23125d5-6fb5-4e7c-a461-189c03528756%2F9ik9qqp_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercises 5–6: Solving Systems of Linear Equations from Augmented Matrices**
In each part of Exercises 5–6, find a system of linear equations in the unknowns \( x_1, x_2, x_3, \ldots \) that corresponds to the given augmented matrix.
Make sure to express each equation clearly and identify the coefficients for the respective variables as depicted by the matrix.
**Example:**
Given the augmented matrix
\[
\begin{bmatrix}
1 & 2 & 3 & | & 4 \\
0 & 5 & 6 & | & 7
\end{bmatrix}
\]
The corresponding system of linear equations is:
\[
\begin{cases}
x_1 + 2x_2 + 3x_3 = 4 \\
5x_2 + 6x_3 = 7
\end{cases}
\]
Continue this process for each provided matrix, deciphering and writing out the respective equations. This practice will enhance your understanding of converting between matrix representations and standard linear equations.
![### Example 6:
#### a.
\[ \begin{bmatrix}
0 & 3 & -1 & -1 & -1\\
5 & 2 & 0 & -3 & -6
\end{bmatrix} \]
#### b.
\[ \begin{bmatrix}
3 & 0 & 1 & -4 & 3\\
-4 & 0 & 4 & 1 & -3\\
-1 & 3 & 0 & -2 & -9\\
0 & 0 & 0 & -1 & -2
\end{bmatrix} \]
---
### Explanation:
These matrices are part of an example in a linear algebra topic, likely focusing on matrix operations such as addition, multiplication, determinants, or systems of linear equations.
- **Matrix a** is a \(2 \times 5\) matrix, indicating it has 2 rows and 5 columns.
- **Matrix b** is a \(4 \times 5\) matrix, indicating it has 4 rows and 5 columns.
This example can be used to illustrate various operations or properties of matrices such as row reduction, the concept of rank, or solving systems of linear equations using row transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2Ff23125d5-6fb5-4e7c-a461-189c03528756%2Fbxd6t7_processed.png&w=3840&q=75)
Transcribed Image Text:### Example 6:
#### a.
\[ \begin{bmatrix}
0 & 3 & -1 & -1 & -1\\
5 & 2 & 0 & -3 & -6
\end{bmatrix} \]
#### b.
\[ \begin{bmatrix}
3 & 0 & 1 & -4 & 3\\
-4 & 0 & 4 & 1 & -3\\
-1 & 3 & 0 & -2 & -9\\
0 & 0 & 0 & -1 & -2
\end{bmatrix} \]
---
### Explanation:
These matrices are part of an example in a linear algebra topic, likely focusing on matrix operations such as addition, multiplication, determinants, or systems of linear equations.
- **Matrix a** is a \(2 \times 5\) matrix, indicating it has 2 rows and 5 columns.
- **Matrix b** is a \(4 \times 5\) matrix, indicating it has 4 rows and 5 columns.
This example can be used to illustrate various operations or properties of matrices such as row reduction, the concept of rank, or solving systems of linear equations using row transformations.
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