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This question involves analyzing the augmented matrix for a system of linear equations. Let's examine the matrix given:

\[
\begin{bmatrix} 
1 & -1 & 0 & 0 & | & 5 \\
0 & 1 & 0 & 0 & | & 6 \\
0 & 1 & -1 & 1 & | & 3 \\
0 & 0 & 1 & -1 & | & 4 \\
\end{bmatrix}
\]

Each row of this matrix represents a linear equation with four unknowns (\( x_1, x_2, x_3, x_4 \)).

The augmented matrix can be broken down into equations:
1. \( x_1 - x_2 = 5 \)
2. \( x_2 = 6 \)
3. \( x_2 - x_3 + x_4 = 3 \)
4. \( x_3 - x_4 = 4 \)

### Steps to Analyze the Solution
1. **Substitute \( x_2 = 6 \) from the second equation** into the others:
   - In the first equation, \( x_1 - 6 = 5 \Rightarrow x_1 = 11 \).
   
2. **Using \( x_2 = 6 \) in the third equation**:
   - \( 6 - x_3 + x_4 = 3 \Rightarrow -x_3 + x_4 = -3 \Rightarrow x_3 - x_4 = 3 \).

3. **Compare with the fourth equation**: \( x_3 - x_4 = 4 \).  
   - This results in a contradiction because \( x_3 - x_4 \) cannot be both 3 and 4 at the same time.

### Conclusion
Due to the contradiction, this system of equations **has no solutions**.

### Answer
The correct answer is:
**A. It has no solutions.**

Would you like more details on any part of this solution?

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Here are some related questions:
1. What is an augmented matrix and how is it used to represent systems of linear equations?
2. How do you determine if a system has no solution, a unique solution, or infinitely many solutions?
3. What are the steps to row reduce an augmented matrix to check for consistency?
4. How do contradictions in a system of equations lead to no solutions?
5. Can there be a case where two equations represent the same line in such systems?

**Tip:** Contradictions in the equations of a system indicate inconsistency, leading to no solutions.

Consider the following augmented matrix. What can you say about the system of equations represented by this augmented matrix? A. It has no solutions. B. It has a unique solution. C. It has an infinite number of solutions. D. None of the above.

Analyze and solve a system of linear equations represented by an augmented matrix. Learn to determine if the system has no solutions, a unique solution, or infinitely many solutions. This problem covers key concepts in linear algebra and matrix consistency, suitable for Grades 10-12.

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