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Let's solve the system of linear equations from the augmented matrix provided:

The augmented matrix is:

\[
\begin{bmatrix}
1 & -7 & 0 & -1 & 0 & -9 \\
0 & 1 & 0 & 0 & 3 & 7 \\
0 & 0 & 1 & 0 & 8 & 9 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

This represents the following system of equations:

1. \( x_1 - 7x_2 - x_4 = -9 \)
2. \( x_2 + 3x_5 = 7 \)
3. \( x_3 + 8x_5 = 9 \)

We need to solve this system for \( x_1, x_2, x_3, x_4, x_5 \).

### Step 1: Identify the pivot variables
From the matrix, we can identify that:
- \( x_1 \), \( x_2 \), and \( x_3 \) are **basic** variables (they have leading 1's in their columns).
- \( x_4 \) and \( x_5 \) are **free** variables (they do not have leading 1's).

### Step 2: Express basic variables in terms of free variables
- From equation (1):  
  \( x_1 = 7x_2 + x_4 - 9 \)

- From equation (2):  
  \( x_2 = 7 - 3x_5 \)

- From equation (3):  
  \( x_3 = 9 - 8x_5 \)

### Step 3: General solution
- \( x_1 = 7(7 - 3x_5) + x_4 - 9 = 49 - 21x_5 + x_4 - 9 = 40 - 21x_5 + x_4 \)
- \( x_2 = 7 - 3x_5 \)
- \( x_3 = 9 - 8x_5 \)
- \( x_4 \) and \( x_5 \) are free.

### Conclusion
The correct solution corresponds to **option A**, where:
- \( x_1 = 40 - 21x_5 + x_4 \)
- \( x_2 = 7 - 3x_5 \)
- \( x_3 = 9 - 8x_5 \)
- \( x_4 \) is free
- \( x_5 \) is free

Would you like further details on any of the steps?

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Here are 5 related questions for you to consider:
1. How do you determine free and basic variables from a matrix?
2. What does it mean for a system to be inconsistent?
3. Can you always express solutions in terms of free variables?
4. What happens if all variables are free?
5. How does row-reduction help in solving a system of linear equations?

**Tip**: When solving augmented matrix problems, always check if the last row corresponds to a valid equation. If it results in something like \( 0 = 1 \), the system is inconsistent.

Find the general solution of the system whose augmented matrix is given below:

1  -7  0  -1  0  -9
0  1  0  0  3  7
0  0  1  0  8  9
0  0  0  0  0  0

This page explains how to solve a system of linear equations using an augmented matrix. We reduce the matrix to echelon form and express basic variables in terms of free variables. Explore the steps with a detailed solution and a breakdown of the process using Gaussian elimination. Suitable for advanced high school or undergraduate students.

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