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Let's break down the problem from the image you uploaded:

### Problem Breakdown:
1. **System of Equations Given:**
   \[
   \begin{aligned}
   2x - y - 2z &= 4, \\
   3x + y + z &= 5, \\
   4x + 2y - z &= 6.
   \end{aligned}
   \]

2. **Create the Augmented Matrix:**
   The augmented matrix for the system of equations is:
   \[
   \begin{bmatrix}
   2 & -1 & -2 & \vert & 4 \\
   3 & 1 & 1 & \vert & 5 \\
   4 & 2 & -1 & \vert & 6
   \end{bmatrix}
   \]

3. **Convert to Row-Echelon Form (REF):**
   The row-echelon form (REF) involves using Gaussian elimination. The goal is to get a matrix where the left part forms an upper triangular matrix.

   After performing the necessary row operations:
   \[
   \begin{bmatrix}
   2 & -1 & -2 & \vert & 4 \\
   0 & 5 & 5 & \vert & -1 \\
   0 & 0 & -10 & \vert & -10
   \end{bmatrix}
   \]

4. **Interpret the Results:**
   The next steps involve identifying whether the system is consistent or inconsistent, dependent or independent.

### Corresponding Simplified System:
The simplified matrix corresponds to the system:
\[
\begin{aligned}
2x - y - 2z &= 4, \\
5y + 5z &= -1, \\
-10z &= -10.
\end{aligned}
\]

### Solving the System:
- From the third equation: \( z = 1 \).
- Substitute \( z = 1 \) into the second equation to find \( y \):
  \[
  5y + 5(1) = -1 \Rightarrow 5y = -6 \Rightarrow y = -\frac{6}{5}.
  \]
- Substitute \( y = -\frac{6}{5} \) and \( z = 1 \) into the first equation to find \( x \):
  \[
  2x - \left(-\frac{6}{5}\right) - 2(1) = 4 \Rightarrow 2x + \frac{6}{5} - 2 = 4.
  \]
  Simplifying:
  \[
  2x = 4 - \frac{6}{5} + 2 = \frac{18}{5} \Rightarrow x = \frac{9}{5}.
  \]

### Final Answer:
The solution to the system is \( x = \frac{9}{5} \), \( y = -\frac{6}{5} \), and \( z = 1 \).

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Would you like more detailed steps for the row operations, or do you have any other questions? Here are five related questions to deepen your understanding:

1. How do you determine if a system of equations is consistent or inconsistent?
2. What are the conditions under which a system is dependent or independent?
3. Can you explain how to perform row operations to achieve the row-echelon form?
4. What is the difference between row-echelon form and reduced row-echelon form?
5. How do you interpret a solution set when a system has infinitely many solutions?

**Tip:** When performing row operations, it's essential to keep track of the operations to avoid mistakes and easily backtrack if needed.

Consider the following linear system: 2x - y - 2z = 4, 3x + y + z = 5, 4x + 2y - z = 6. Create the augmented matrix of this system. Convert the system into its row-echelon form and solve it.

Learn how to solve a system of linear equations by converting it into row-echelon form using Gaussian elimination. This problem involves creating an augmented matrix and performing row operations to simplify and solve the system. Suitable for students in Grades 10-12.

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