https://file.aiquickdraw.com/mathbot/file/1726077960641vc8pyyz2.jpg

The system of equations from the image is:

\[
\begin{aligned}
1) \quad x - 2y + z &= 9 \\
2) \quad 2x + y - z &= 3 \\
3) \quad -2x + 4y - 2z &= 2
\end{aligned}
\]

We can solve this system using any method, such as substitution, elimination, or matrices (Gaussian elimination). I will use **Gaussian elimination** to find the solution.

### Step 1: Write the augmented matrix for the system.

\[
\begin{bmatrix}
1 & -2 & 1 & | & 9 \\
2 & 1 & -1 & | & 3 \\
-2 & 4 & -2 & | & 2
\end{bmatrix}
\]

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

- First, use \( R_1 \) to eliminate \( x \) from \( R_2 \) and \( R_3 \).

\[
R_2 \leftarrow R_2 - 2R_1 \quad \text{and} \quad R_3 \leftarrow R_3 + 2R_1
\]

\[
\begin{bmatrix}
1 & -2 & 1 & | & 9 \\
0 & 5 & -3 & | & -15 \\
0 & 0 & 0 & | & 20
\end{bmatrix}
\]

### Step 3: Analyze the system.

The last row \( [0 \quad 0 \quad 0 | 20] \) indicates an **inconsistent system**, meaning there is no solution. This is because it results in the equation \( 0 = 20 \), which is a contradiction.

### Conclusion:
The system of equations has **no solution**.

Would you like further details on this? Feel free to ask!

Here are five related questions for further practice:
1. How do you determine if a system of equations has no solution?
2. What are the differences between inconsistent, consistent, and dependent systems?
3. How can Gaussian elimination be used to find the inverse of a matrix?
4. What is the geometrical interpretation of a system of equations with no solution?
5. How can substitution or elimination methods show an inconsistent system?

**Tip:** Always check for contradictions in augmented matrices when using Gaussian elimination to solve systems of equations!

Solve the system of equations: x - 2y + z = 9, 2x + y - z = 3, -2x + 4y - 2z = 2

Explore how to solve the system of equations: x - 2y + z = 9, 2x + y - z = 3, -2x + 4y - 2z = 2 using Gaussian elimination. Learn key concepts in algebra and understand why the system is inconsistent. Suitable for Grades 10-12.

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