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?

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.