https://file.aiquickdraw.com/mathbot/file/1727048204614rgqqg3in.jpg

The image shows a system of three linear equations:

\[
\begin{aligned}
4x_1 - x_2 + x_3 &= -12, \\
2x_1 + 2x_2 + 3x_3 &= 9, \\
5x_1 - 2x_2 + 6x_3 &= 8.
\end{aligned}
\]

The question asks to solve this system using **Cramer's Rule**. Based on the final answer shown in the image \(\left( \frac{-35}{9}, \frac{11}{9}, \frac{42}{9} \right)\), the result was marked incorrect. Let's check why this might be.

### Steps to solve using Cramer's Rule:

1. **Find the determinant of the coefficient matrix** (\(D\)):

\[
A = \begin{pmatrix}
4 & -1 & 1 \\
2 & 2 & 3 \\
5 & -2 & 6
\end{pmatrix}
\]

2. **Calculate determinants of modified matrices**:
   - \(D_{x_1}\), \(D_{x_2}\), and \(D_{x_3}\) by replacing the respective column in the matrix with the constants \([-12, 9, 8]\).

3. **Solve for** \(x_1\), \(x_2\), and \(x_3\) using:

\[
x_1 = \frac{D_{x_1}}{D}, \quad x_2 = \frac{D_{x_2}}{D}, \quad x_3 = \frac{D_{x_3}}{D}.
\]

Let me calculate this step by step to see what the correct solution should be.The determinant of the coefficient matrix \( A \) is approximately \( 55 \). Using Cramer's Rule, the solutions to the system of equations are:

\[
x_1 = -4, \quad x_2 = 1, \quad x_3 = 5.
\]

It seems that the solution provided in the image \(\left( \frac{-35}{9}, \frac{11}{9}, \frac{42}{9} \right)\) is incorrect, and the correct values for \( x_1, x_2, x_3 \) are \( -4 \), \( 1 \), and \( 5 \), respectively.

Would you like more details or have any questions about this?

Here are five questions related to this problem:
1. How is Cramer's Rule applied in 3x3 systems?
2. What happens if the determinant \( D \) of the coefficient matrix is zero?
3. How do you calculate the determinant of a 3x3 matrix?
4. How is Cramer's Rule different from other methods like Gaussian elimination?
5. Can Cramer's Rule be applied to larger systems, such as 4x4 matrices?

**Tip:** When using Cramer's Rule, always check the determinant of the coefficient matrix first. If it's zero, the system either has no solution or infinitely many solutions.

Use Cramer's Rule to solve (if possible) the system of linear equations: 4x1 - x2 + x3 = -12, 2x1 + 2x2 + 3x3 = 9, 5x1 - 2x2 + 6x3 = 8.

This page provides a detailed solution using Cramer's Rule for the system of linear equations: 4x1 - x2 + x3 = -12, 2x1 + 2x2 + 3x3 = 9, and 5x1 - 2x2 + 6x3 = 8. The problem demonstrates key concepts in linear algebra, including matrix determinants and Cramer's Rule.

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