https://file.aiquickdraw.com/mathbot/file/1726329683979vh0fcg7j.jpg

The image provides a linear system of equations along with its reduced row echelon form (RREF). The system to solve is:

\[
\begin{aligned}
-x + y + 2z &= 2, \\
x - 2y - z &= -6, \\
-x + y + 2z &= 2.
\end{aligned}
\]

The RREF of the augmented matrix for this system is:

\[
\begin{bmatrix}
1 & 0 & -3 & 2 \\
0 & 1 & -1 & 4 \\
0 & 0 & 0 & 0
\end{bmatrix}.
\]

### Step 1: Interpret the RREF
The RREF gives us the following system of equations:

\[
\begin{aligned}
x - 3z &= 2, \\
y - z &= 4.
\end{aligned}
\]

### Step 2: Solve for \(x\) and \(y\)
- From \(x - 3z = 2\), we solve for \(x\):
  \[
  x = 3z + 2.
  \]
  
- From \(y - z = 4\), we solve for \(y\):
  \[
  y = z + 4.
  \]

### Step 3: Parametrize the Solution
Since \(z\) is a free variable, the general solution can be written in terms of \(z\):
\[
\begin{aligned}
x &= 3z + 2, \\
y &= z + 4, \\
z &= z.
\end{aligned}
\]

Thus, the parametric solution for the system is:
\[
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3z + 2 \\ z + 4 \\ z \end{bmatrix}.
\]

### Summary of Solution:
The solution is parametrized by \(z\), and the general form is:
\[
x = 3z + 2, \quad y = z + 4, \quad z = z.
\]

Let me know if you need any further clarification!

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Here are some related questions you might find useful:
1. How do you compute the RREF of a matrix?
2. What are free variables, and why are they important in systems of equations?
3. How do you interpret the results from an inconsistent system?
4. Can every system of linear equations be solved using RREF?
5. How do you use matrix operations to solve a system of equations?

**Tip:** When solving systems of equations, always check if the system is consistent by looking at the last row of the RREF matrix!

Use the reduced row echelon form above to solve the system

Learn how to solve a system of linear equations using the reduced row echelon form (RREF). This guide provides step-by-step solutions, explaining how to parameterize the answer using free variables in the system, suitable for college-level students.

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