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To solve the given system of linear equations, we will proceed by writing the system in matrix form and then performing row operations.

The system is:
\[
\begin{cases}
x_1 + x_2 - 2x_3 = -1, \\
3x_1 + 2x_2 - 2x_3 = 2.
\end{cases}
\]

### Step 1: Convert the System to an Augmented Matrix
The augmented matrix for this system is:
\[
\begin{bmatrix}
1 & 1 & -2 & | & -1 \\
3 & 2 & -2 & | & 2 \\
\end{bmatrix}
\]

### Step 2: Perform Row Operations
1. Subtract 3 times the first row from the second row to eliminate \( x_1 \) from the second equation:
   \[
   R_2 = R_2 - 3 \times R_1
   \]
   This gives:
   \[
   \begin{bmatrix}
   1 & 1 & -2 & | & -1 \\
   0 & -1 & 4 & | & 5 \\
   \end{bmatrix}
   \]

2. Now, divide the second row by \(-1\) to simplify:
   \[
   R_2 = -1 \times R_2
   \]
   This gives:
   \[
   \begin{bmatrix}
   1 & 1 & -2 & | & -1 \\
   0 & 1 & -4 & | & -5 \\
   \end{bmatrix}
   \]

3. Finally, subtract the second row from the first row to eliminate \( x_2 \) from the first equation:
   \[
   R_1 = R_1 - R_2
   \]
   This results in:
   \[
   \begin{bmatrix}
   1 & 0 & 2 & | & 4 \\
   0 & 1 & -4 & | & -5 \\
   \end{bmatrix}
   \]

### Step 3: Rewrite the System
Now, the system of equations is:
\[
\begin{cases}
x_1 + 2x_3 = 4, \\
x_2 - 4x_3 = -5.
\end{cases}
\]

### Step 4: Express in Terms of \( t \) (Free Variable)
Let \( x_3 = t \), where \( t \) is a free variable. Then:
1. From the first equation: \( x_1 = 4 - 2t \).
2. From the second equation: \( x_2 = -5 + 4t \).

### Solution
The solution in vector form is:
\[
\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4 \\ -5 \\ 0 \end{bmatrix} + t \begin{bmatrix} -2 \\ 4 \\ 1 \end{bmatrix},
\]
where \( t \) is a free variable.

This is the complete solution to the system.

---

Would you like further details or have any questions?

### Related Questions:
1. How do you identify free variables in a system of linear equations?
2. What are the general steps to solve a system of equations by the matrix method?
3. How does row reduction help in finding solutions for systems with free variables?
4. What does it mean if a system has infinitely many solutions?
5. How can you verify the solution for a system of linear equations?

### Tip:
When solving systems with free variables, remember that each free variable corresponds to a degree of freedom in the solution set, leading to infinitely many solutions.

Solve the system: { x1 + x2 - 2x3 = -1, 3x1 + 2x2 - 2x3 = 2 }

Learn how to solve a system of linear equations with a free variable using matrix row reduction. This example covers key concepts in linear algebra, including matrix operations and Gaussian elimination. Suitable for students in Grades 10-12.

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