To solve a system of linear equations using the row echelon form, follow these steps:

### 1. Write the Augmented Matrix:
Given a system of linear equations, express it as an augmented matrix.

For example, consider the system:
\[
\begin{aligned}
x + 2y + 3z &= 9 \\
2x + 3y + z &= 8 \\
3x + y + 2z &= 7
\end{aligned}
\]

The augmented matrix is:
\[
\begin{pmatrix}
1 & 2 & 3 & | & 9 \\
2 & 3 & 1 & | & 8 \\
3 & 1 & 2 & | & 7
\end{pmatrix}
\]

### 2. Perform Row Operations to Achieve Row Echelon Form:
The goal is to obtain a triangular form (where all elements below the main diagonal are zero).

**Step 1: Make the leading entry of the first row 1** (which is already done here).

**Step 2: Eliminate the entries below the first leading 1.** Subtract appropriate multiples of the first row from the rows below.

- \( R_2 \rightarrow R_2 - 2R_1 \) 
- \( R_3 \rightarrow R_3 - 3R_1 \)

This results in:
\[
\begin{pmatrix}
1 & 2 & 3 & | & 9 \\
0 & -1 & -5 & | & -10 \\
0 & -5 & -7 & | & -20
\end{pmatrix}
\]

**Step 3: Make the leading entry of the second row 1.** Multiply the second row by \(-1\).

\[
\begin{pmatrix}
1 & 2 & 3 & | & 9 \\
0 & 1 & 5 & | & 10 \\
0 & -5 & -7 & | & -20
\end{pmatrix}
\]

**Step 4: Eliminate the entries below the second leading 1.** Add appropriate multiples of the second row to the rows below.

- \( R_3 \rightarrow R_3 + 5R_2 \)

This results in:
\[
\begin{pmatrix}
1 & 2 & 3 & | & 9 \\
0 & 1 & 5 & | & 10 \\
0 & 0 & 18 & | & 30
\end{pmatrix}
\]

Now, the matrix is in row echelon form.

### 3. Solve the System by Back-Substitution:
Start from the bottom row and work your way upwards.

- From the third row: \( 18z = 30 \) → \( z = \frac{30}{18} = \frac{5}{3} \)
- From the second row: \( y + 5z = 10 \) → \( y + 5\left(\frac{5}{3}\right) = 10 \) → \( y = 10 - \frac{25}{3} = \frac{30 - 25}{3} = \frac{5}{3} \)
- From the first row: \( x + 2y + 3z = 9 \) → \( x + 2\left(\frac{5}{3}\right) + 3\left(\frac{5}{3}\right) = 9 \) → \( x = 9 - \frac{10}{3} - 5 = \frac{27 - 10 - 15}{3} = \frac{2}{3} \)

### Final Solution:
\[
x = \frac{2}{3}, \quad y = \frac{5}{3}, \quad z = \frac{5}{3}
\]

### Would you like further details on any step, or do you have any questions?

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Here are 8 related questions that expand on this topic:

1. How do you perform Gaussian elimination to solve a system of equations?
2. What is the difference between row echelon form and reduced row echelon form?
3. Can you explain how to interpret the solutions of a system from its echelon form?
4. How do you determine if a system has no solution, one solution, or infinitely many solutions using row echelon form?
5. What are the advantages of using matrix operations over substitution or elimination methods in solving linear systems?
6. How can row echelon form be used in solving systems with more than three variables?
7. What is the role of pivot elements in row echelon form?
8. How does the concept of linear independence relate to row echelon form?

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**Tip:** When performing row operations, always aim to simplify each step as much as possible to avoid unnecessary fractions or complex calculations.

Learn how to solve a system of linear equations using the row echelon form with detailed steps and examples. This tutorial covers the process of converting a system of equations into its augmented matrix form, performing row operations to achieve row echelon form, and solving the system using back-substitution. Suitable for college-level students studying linear algebra and systems of equations.

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