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We are given an augmented matrix of a system of linear equations:

\[
\begin{bmatrix}
1 & -1 & -1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 1 & 1 & 1
\end{bmatrix}
\]

To determine the nature of the solution, we will perform **row reduction** (Gaussian elimination) to transform this matrix into row echelon form. This will help us identify the number of free variables or whether the system has a unique solution or no solutions.

Let's begin by working on the row reduction. 

Starting with the matrix:

\[
\begin{bmatrix}
1 & -1 & -1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 1 & 1 & 1
\end{bmatrix}
\]

Notice that rows 2 and 3 are identical. Subtracting Row 2 from Row 3:

\[
R_3 \rightarrow R_3 - R_2
\]

Results in:

\[
\begin{bmatrix}
1 & -1 & -1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

Now that the matrix is in **row echelon form**, let's analyze the system.

- The third row is all zeros, indicating that one equation is redundant.
- The first and second rows provide two independent equations with three unknowns, implying **one free variable**.

Thus, the system has infinitely many solutions with **one free variable**.

### Conclusion:
The correct answer is **(a) infinitely many solutions with one free variable**.

---

Would you like further clarification or details on the row reduction process?

Here are 5 related questions for deeper understanding:
1. How do we identify free variables in a system of equations?
2. What is the significance of a row of zeros in an augmented matrix?
3. How does Gaussian elimination help solve systems of equations?
4. Can a system with more unknowns than equations have a unique solution?
5. What are the possible solution types for a system of linear equations?

**Tip:** Always check for row redundancy when solving systems of equations—it can simplify the problem by reducing the number of independent equations.

Consider the following augmented matrix of a system of linear equations:
[ 1  -1  -1  1 ]
[ 0   1   1  1 ]
[ 0   1   1  1 ]

The system has:
(a) infinitely many solutions with one free variable
(b) infinitely many solutions with two free variables
(c) unique solution
(d) no solutions

Learn how to analyze a system of linear equations represented by an augmented matrix using Gaussian elimination. Explore the process of determining the number of free variables and whether the system has infinitely many solutions or a unique solution.

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