Below are augmented matrices (in reduced row echelon form) corresponding to the systems of linear equations. Determine if each system is consistent with a unique solution, consistent with infinitely many solutions or inconsistent: [(1,0,-9,2),(0,1,-1,-6),(0,0,0,0)]

Let's analyze the augmented matrix:

\[
\begin{pmatrix}
1 & 0 & -9 & 2 \\
0 & 1 & -1 & -6 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

This is the reduced row echelon form (RREF) of the augmented matrix. The matrix represents the following system of equations:

1. \( x_1 - 9x_3 = 2 \)
2. \( x_2 - x_3 = -6 \)
3. \( 0 = 0 \) (which is always true and doesn't add any new information)

### Step 1: Analyze the system
- There are three unknowns: \( x_1 \), \( x_2 \), and \( x_3 \).
- There are two non-trivial equations (from the first two rows), while the third row gives \( 0 = 0 \), which is trivially true and doesn't affect the solution space.

### Step 2: Determine if the system is consistent
- The system is consistent because there is no row that corresponds to an equation like \( 0 = c \), where \( c \) is a non-zero constant (which would indicate inconsistency).
- The presence of a free variable (since \( x_3 \) is not a leading variable) suggests that there are infinitely many solutions.

### Step 3: Identify the type of solution
- \( x_3 \) is a free variable because it does not have a leading 1 in the matrix, meaning we can assign any value to \( x_3 \).
- We can express \( x_1 \) and \( x_2 \) in terms of \( x_3 \).

Thus, the system has **infinitely many solutions**, as there is at least one free variable (in this case, \( x_3 \)).

### Conclusion:
The system is consistent with infinitely many solutions.

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Would you like details on how to parametrize the solutions or any further explanation?

Here are 5 related questions:
1. How do you identify free variables in a system of equations?
2. What makes a system of equations inconsistent?
3. How do you find the general solution of a system with infinitely many solutions?
4. What is the difference between row echelon form and reduced row echelon form?
5. Can a system have both a unique and infinite solution?

**Tip:** In RREF, any row that is completely zero indicates that the system has either infinitely many solutions or no solution, depending on other rows.

Analyze the augmented matrix in reduced row echelon form and determine whether the system of linear equations is consistent. Understand the concept of free variables and learn how the system leads to infinitely many solutions. Suitable for advanced high school or college-level students.

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