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Let's solve this system of linear equations by both the **Elementary Row Operation method** and **Gaussian Elimination**.

### Given System of Equations
1. \( x - 2y - 4z = 4 \)
2. \( 2x + y + z = 9 \)
3. \( x + y - z = 1 \)

### Step 1: Elementary Row Operations

We'll use the augmented matrix form of the system:

\[
\begin{pmatrix} 1 & -2 & -4 & | & 4 \\ 2 & 1 & 1 & | & 9 \\ 1 & 1 & -1 & | & 1 \end{pmatrix}
\]

Perform row operations to bring this matrix to row-echelon form:

1. **Make the first entry of the first row (pivot) 1** (it already is).
2. **Eliminate the first entry of rows 2 and 3 using Row 1.**
   - Row 2: \( R_2 - 2R_1 \)
   - Row 3: \( R_3 - R_1 \)

   After performing these operations, we get:

\[
\begin{pmatrix} 1 & -2 & -4 & | & 4 \\ 0 & 5 & 9 & | & 1 \\ 0 & 3 & 3 & | & -3 \end{pmatrix}
\]

3. **Make the pivot in the second row 1** by dividing \( R_2 \) by 5:

\[
\begin{pmatrix} 1 & -2 & -4 & | & 4 \\ 0 & 1 & \frac{9}{5} & | & \frac{1}{5} \\ 0 & 3 & 3 & | & -3 \end{pmatrix}
\]

4. **Eliminate the second entry of Row 3 using Row 2**:
   - Row 3: \( R_3 - 3 \cdot R_2 \)

   After this operation:

\[
\begin{pmatrix} 1 & -2 & -4 & | & 4 \\ 0 & 1 & \frac{9}{5} & | & \frac{1}{5} \\ 0 & 0 & -\frac{24}{5} & | & -\frac{18}{5} \end{pmatrix}
\]

5. **Make the pivot in the third row 1** by dividing \( R_3 \) by \(-\frac{24}{5}\):

\[
\begin{pmatrix} 1 & -2 & -4 & | & 4 \\ 0 & 1 & \frac{9}{5} & | & \frac{1}{5} \\ 0 & 0 & 1 & | & \frac{3}{4} \end{pmatrix}
\]

6. **Back-substitute to eliminate the entries above the pivot in Row 3**:
   - Update Row 2 and Row 1 to eliminate the \( z \)-terms.

This gives the final matrix, from which we can read off the values of \( x \), \( y \), and \( z \).

Would you like a detailed solution for **Gaussian Elimination**, or is this summary sufficient? 

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Here are five additional related questions for further exploration:

1. How does Gaussian Elimination differ from Gauss-Jordan Elimination?
2. Can we apply these methods to systems with more than three variables?
3. What conditions must be met for a system of equations to have a unique solution?
4. How do we determine if a system has infinitely many solutions or is inconsistent?
5. How does the matrix determinant help us understand the nature of solutions?

**Tip:** Elementary Row Operations and Gaussian Elimination are essential tools for solving systems of equations efficiently, especially in linear algebra applications.

Given the following system of linear equations:

x - 2y - 4z = 4
2x + y + z = 9
x + y - z = 1

Find the value of x, y, and z using the Elementary Row Operation method and Gaussian Elimination.

Learn how to solve a system of linear equations using the Elementary Row Operation method and Gaussian Elimination. This example includes the system: x - 2y - 4z = 4, 2x + y + z = 9, and x + y - z = 1. Perfect for students in advanced high school algebra or linear algebra courses.

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