https://file.aiquickdraw.com/mathbot/file/1725340645542bkyiw3or.jpg

The system of linear equations shown in the image is:

\[
\begin{aligned}
1) \quad & x - 4y - z = 11 \\
2) \quad & 2x - 5y + 2z = 39 \\
3) \quad & -3x + 2y + z = 1
\end{aligned}
\]

To solve this system using Cramer's rule, we need to:

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

\[
D = \begin{vmatrix}
1 & -4 & -1 \\
2 & -5 & 2 \\
-3 & 2 & 1
\end{vmatrix}
\]

2. **Find the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constants from the right side of the equations (Dx, Dy, Dz):**

- For \( Dx \), replace the first column with the constants:
\[
Dx = \begin{vmatrix}
11 & -4 & -1 \\
39 & -5 & 2 \\
1 & 2 & 1
\end{vmatrix}
\]

- For \( Dy \), replace the second column with the constants:
\[
Dy = \begin{vmatrix}
1 & 11 & -1 \\
2 & 39 & 2 \\
-3 & 1 & 1
\end{vmatrix}
\]

- For \( Dz \), replace the third column with the constants:
\[
Dz = \begin{vmatrix}
1 & -4 & 11 \\
2 & -5 & 39 \\
-3 & 2 & 1
\end{vmatrix}
\]

3. **Calculate \( x \), \( y \), and \( z \) using the formulas:**
\[
x = \frac{Dx}{D}, \quad y = \frac{Dy}{D}, \quad z = \frac{Dz}{D}
\]

Let's compute these determinants and solve for \( x \), \( y \), and \( z \).It seems there was a timeout error while trying to compute the solution. Let me try that again to calculate the values of \( x \), \( y \), and \( z \).It seems there was a small issue with the code, specifically an undefined module. Let me correct that and try again.The solutions for the system of equations using Cramer's rule are:

\[
x = -1, \quad y = -5, \quad z = 8
\]

These values satisfy all three equations given in the problem.

Would you like further details or have any questions? Here are some related topics you might explore:

1. How does Cramer's rule apply to systems of more than three equations?
2. What are the conditions for using Cramer's rule?
3. How would you solve this system using the inverse matrix method?
4. What are some limitations of Cramer's rule in solving linear systems?
5. Can you derive the determinant formula used in Cramer's rule?

**Tip:** When solving systems of equations, always check if the determinant of the coefficient matrix is non-zero before applying Cramer's rule, as it won't work if the determinant is zero.

Learn how to solve a system of linear equations using Cramer's Rule with a detailed step-by-step solution. This problem involves calculating determinants and applying Cramer's Rule to find the values of x, y, and z. Suitable for advanced high school students.

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