https://r2.erweima.ai/mathbot/image/1722834902389v2i41cxb.jpg

The text extraction from the image was not successful. However, I can try to interpret and solve the problem visually.

From the image, it appears the question asks to "Convert the system of equations to a system of equations using basic variables (\( x_1, x_2, \ldots \))."

The system of equations seems to be:
\[ x_1 - 2x_2 + 3x_3 = 5 \]
\[ 2x_1 + x_2 - x_3 = 3 \]
\[ -x_1 + 3x_2 + 2x_3 = -1 \]

Let's rewrite and solve this system:

1. **Write the augmented matrix for the system:**
\[
\left[\begin{array}{ccc|c}
1 & -2 & 3 & 5 \\
2 & 1 & -1 & 3 \\
-1 & 3 & 2 & -1
\end{array}\right]
\]

2. **Perform row operations to get the matrix in row-echelon form:**

\[ R_2 \leftarrow R_2 - 2R_1 \]
\[ R_3 \leftarrow R_3 + R_1 \]

\[
\left[\begin{array}{ccc|c}
1 & -2 & 3 & 5 \\
0 & 5 & -7 & -7 \\
0 & 1 & 5 & 4
\end{array}\right]
\]

3. **Further simplify:**

\[ R_3 \leftarrow R_3 - \frac{1}{5} R_2 \]

\[
\left[\begin{array}{ccc|c}
1 & -2 & 3 & 5 \\
0 & 5 & -7 & -7 \\
0 & 0 & \frac{28}{5} & \frac{31}{5}
\end{array}\right]
\]

4. **Convert back to system of equations:**

\[
\begin{cases}
x_1 - 2x_2 + 3x_3 = 5 \\
5x_2 - 7x_3 = -7 \\
\frac{28}{5}x_3 = \frac{31}{5}
\end{cases}
\]

5. **Solve for \( x_3 \):**

\[
x_3 = \frac{31}{28}
\]

6. **Substitute \( x_3 \) into the second equation to find \( x_2 \):**

\[
5x_2 - 7\left(\frac{31}{28}\right) = -7
\]

Solving for \( x_2 \):

\[
5x_2 - \frac{217}{28} = -7
\]
\[
5x_2 = -7 + \frac{217}{28}
\]
\[
5x_2 = -7 + \frac{217}{28}
\]
\[
5x_2 = -7 + 7.75
\]
\[
5x_2 = 0.75
\]
\[
x_2 = \frac{0.75}{5}
\]
\[
x_2 = 0.15
\]

7. **Substitute \( x_2 \) and \( x_3 \) into the first equation to find \( x_1 \):**

\[
x_1 - 2(0.15) + 3\left(\frac{31}{28}\right) = 5
\]
\[
x_1 - 0.3 + \frac{93}{28} = 5
\]
\[
x_1 - 0.3 + 3.32 = 5
\]
\[
x_1 + 3.02 = 5
\]
\[
x_1 = 1.98
\]

Thus, the solutions are:
\[
x_1 = 1.98, \quad x_2 = 0.15, \quad x_3 = \frac{31}{28}
\]

Would you like more details or have any questions? Here are some relative questions to expand on this:

1. What are row operations and why are they used in solving systems of equations?
2. How do you determine which row operations to perform?
3. What is the difference between row-echelon form and reduced row-echelon form?
4. How can you verify the solutions of a system of equations?
5. What are some alternative methods for solving systems of linear equations?
6. How does the method of substitution differ from using augmented matrices?
7. What are basic variables in the context of systems of equations?
8. How do you interpret the solution of a system of equations geometrically?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure they satisfy all equations.

Learn how to convert a system of equations to basic variables and solve it step-by-step. This problem involves linear algebra concepts, including augmented matrices and row operations. Suitable for college-level students studying linear algebra.

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