Show that the equations
x + y + z = - 3
3x + y - 2z = - 2
2x + 4y + 7z = 7
are not consistent using Echelon form.

To show that the given system of equations is not consistent using Echelon form, we need to convert the system into an augmented matrix and then perform row operations to obtain the Echelon form. Here's the step-by-step process:

### Step 1: Write the Augmented Matrix

The system of equations is:
\[ 
\begin{cases}
x + y + z = -3 \\
3x + y - 2z = -2 \\
2x + 4y + 7z = 7
\end{cases}
\]

The corresponding augmented matrix is:
\[
\begin{pmatrix}
1 & 1 & 1 & \vert & -3 \\
3 & 1 & -2 & \vert & -2 \\
2 & 4 & 7 & \vert & 7
\end{pmatrix}
\]

### Step 2: Perform Row Operations

**Step 2.1: Eliminate the \(x\)-terms below the first row**

- Subtract 3 times the first row from the second row:
\[
R2 \leftarrow R2 - 3R1 \\
R2 = (3, 1, -2, -2) - 3(1, 1, 1, -3) \\
R2 = (3 - 3, 1 - 3, -2 - 3, -2 + 9) \\
R2 = (0, -2, -5, 7)
\]

- Subtract 2 times the first row from the third row:
\[
R3 \leftarrow R3 - 2R1 \\
R3 = (2, 4, 7, 7) - 2(1, 1, 1, -3) \\
R3 = (2 - 2, 4 - 2, 7 - 2, 7 + 6) \\
R3 = (0, 2, 5, 13)
\]

The matrix is now:
\[
\begin{pmatrix}
1 & 1 & 1 & \vert & -3 \\
0 & -2 & -5 & \vert & 7 \\
0 & 2 & 5 & \vert & 13
\end{pmatrix}
\]

**Step 2.2: Eliminate the \(y\)-terms below the second row**

- Add the second row to the third row:
\[
R3 \leftarrow R3 + R2 \\
R3 = (0, 2, 5, 13) + (0, -2, -5, 7) \\
R3 = (0, 0, 0, 20)
\]

The matrix is now:
\[
\begin{pmatrix}
1 & 1 & 1 & \vert & -3 \\
0 & -2 & -5 & \vert & 7 \\
0 & 0 & 0 & \vert & 20
\end{pmatrix}
\]

### Step 3: Analyze the Echelon Form

The final row of the matrix:
\[
(0, 0, 0, \vert, 20)
\]
indicates the equation:
\[
0x + 0y + 0z = 20
\]
which simplifies to:
\[
0 = 20
\]

This is a contradiction, indicating that the system of equations is inconsistent.

Thus, we have shown that the given system of equations is not consistent using Echelon form.

Would you like more details or have any questions? Here are some related questions:

1. What is the definition of a consistent system of equations?
2. How do you determine if a matrix is in Row Echelon Form?
3. What are the properties of an inconsistent system of equations?
4. How can you use Gaussian elimination to solve systems of equations?
5. What is the difference between Echelon form and Reduced Row Echelon form?
6. Can a system of linear equations be inconsistent if it has more equations than unknowns?
7. What is the significance of a row of zeros in an augmented matrix?
8. How can you verify the solution to a system of linear equations?

**Tip:** When performing row operations, always aim to create leading ones in each row and then use them to eliminate the variables below. This method helps to systematically reach the Echelon form.

Learn how to demonstrate the inconsistency of a system of linear equations using Echelon form. This problem involves converting equations into an augmented matrix, performing row operations, and analyzing the resulting matrix to prove inconsistency. Suitable for college-level students studying linear algebra.

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