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The given problem involves solving a system of linear equations using the reduced row echelon form (RREF) of a matrix.

The problem provides the system of equations:
\[
\begin{aligned}
-x - y &= -3 \\
-x - 4z &= -4 \\
x - y &= 3
\end{aligned}
\]

We also see the matrix that represents this system and its reduced row echelon form:
\[
\begin{pmatrix}
-1 & -1 & 0 & -3 \\
-1 & 0 & -4 & -4 \\
-1 & -1 & 0 & -3
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & 4 & 4 \\
0 & 1 & -4 & -1 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Step 1: Interpret the RREF Matrix

From the RREF matrix:
\[
\begin{pmatrix}
1 & 0 & 4 & 4 \\
0 & 1 & -4 & -1 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]
we can rewrite this in terms of equations:
1. \( x + 4z = 4 \)
2. \( y - 4z = -1 \)

The third row, being all zeros, indicates that the system has a free variable (in this case, \( z \)).

### Step 2: Solve the System

From the first equation:
\[
x = 4 - 4z
\]
From the second equation:
\[
y = -1 + 4z
\]

### Step 3: General Solution

Since \( z \) is a free variable, we can parametrize the solution in terms of \( z \):
\[
x = 4 - 4z, \quad y = -1 + 4z, \quad z = z
\]

### Final Answer
The parametric solution is:
\[
x = 4 - 4z, \quad y = -1 + 4z, \quad z = z
\]

To enter this into the web form:
- \( x = 4 \)
- \( y = -4 \) 
- \( z = 0 \) (if you want the particular solution where \( z = 0 \)).

Let me know if you need further details, or if you have any questions!

Here are 5 relative questions to expand the problem:
1. How does Gaussian elimination work for solving systems of equations?
2. What is the difference between row echelon form and reduced row echelon form?
3. How do free variables in a system relate to the existence of infinite solutions?
4. How can you verify that the RREF of a matrix is correct?
5. What are some practical applications of solving linear systems?

**Tip:** Always verify your solution by substituting back into the original system to check for consistency.

Given a system of linear equations, use the reduced row echelon form to solve the system. The matrix and its RREF are given.

This problem demonstrates how to solve a system of linear equations by converting a matrix to reduced row echelon form (RREF). It covers key linear algebra concepts, including matrix transformations and handling free variables.

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