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To reduce this matrix to its row echelon form (or reduced row echelon form) using row operations, we start with the augmented matrix:

\[
\begin{bmatrix}
1 & 1 & 1 & | & 18 \\
2 & 3 & 6 & | & 29 \\
\end{bmatrix}
\]

Our goal is to simplify it so that the matrix represents a system of linear equations in its simplest form. Here are the steps:

### Step 1: Make the First Entry in the Second Row Zero
- We want to make the element in the second row, first column (currently 2) equal to 0.
- To do this, we can perform the row operation:
  \[
  \text{Row 2} = \text{Row 2} - 2 \times \text{Row 1}
  \]

Calculating this:
\[
\text{Row 2} = [2, 3, 6, |, 29] - 2 \times [1, 1, 1, |, 18]
\]
\[
= [2 - 2(1), 3 - 2(1), 6 - 2(1), |, 29 - 2(18)]
\]
\[
= [0, 1, 4, |, -7]
\]

Now the matrix is:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 18 \\
0 & 1 & 4 & | & -7 \\
\end{bmatrix}
\]

### Step 2: Make the Second Entry in the First Row Zero
- To simplify further, we can make the element in the first row, second column (currently 1) equal to 0.
- Use the row operation:
  \[
  \text{Row 1} = \text{Row 1} - \text{Row 2}
  \]

Calculating this:
\[
\text{Row 1} = [1, 1, 1, |, 18] - [0, 1, 4, |, -7]
\]
\[
= [1 - 0, 1 - 1, 1 - 4, |, 18 - (-7)]
\]
\[
= [1, 0, -3, |, 25]
\]

So, the matrix now becomes:
\[
\begin{bmatrix}
1 & 0 & -3 & | & 25 \\
0 & 1 & 4 & | & -7 \\
\end{bmatrix}
\]

### Final Matrix in Row-Echelon Form
The matrix is now in row-echelon form:
\[
\begin{bmatrix}
1 & 0 & -3 & | & 25 \\
0 & 1 & 4 & | & -7 \\
\end{bmatrix}
\]

This matrix represents the same system of equations, but in its simplified form.

Would you like further details on the row reduction process, or have any questions?

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**Here are 5 questions that expand on this topic:**

1. What are the key differences between row echelon form and reduced row echelon form?
2. How would you determine if a matrix is already in reduced row echelon form?
3. Can this matrix be further simplified to reduced row echelon form, and if so, how?
4. What are the possible solutions for the system represented by this matrix?
5. How does Gaussian elimination differ from Gauss-Jordan elimination?

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**Tip:** When performing row operations, keep track of each step carefully to avoid calculation errors and ensure that each operation simplifies the matrix closer to the desired form.

Use row operations to change the matrix to reduced form: \[ \begin{bmatrix} 1 & 1 & 1 & | & 18 \\ 2 & 3 & 6 & | & 29 \end{bmatrix} \]

Learn how to use row operations to transform a matrix into its reduced row echelon form. This example covers a 2x3 augmented matrix and applies Gaussian elimination to reach a simplified form, illustrating each step in the process. Suitable for high school students studying linear algebra and matrix transformations.

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