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This image shows the process of using **Cramer's Rule** to solve a system of linear equations for \( z \). The given system of equations is:

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

To solve for \( z \) using Cramer's Rule, the determinant of the coefficient matrix \( D \) is calculated, and the determinant of the modified matrix \( D_z \) is used to find \( z \).

### Steps:
1. **Write the Coefficient Matrix**:
   \[
   A = \begin{pmatrix}
   -2 & 1  & -5 \\
   0  & -1 & 2 \\
   3  & 5  & -2
   \end{pmatrix}
   \]
2. **Calculate the Determinant of the Coefficient Matrix \( D \)**:
   The image shows that the determinant \( D \) is already calculated as \( 7 \).

3. **Form the Matrix \( A_z \) (replace the \( z \)-column with constants from the right side of the equations)**:
   \[
   A_z = \begin{pmatrix}
   -2 & 1 & -2 \\
   0 & -1 & 1 \\
   3 & 5 & -4
   \end{pmatrix}
   \]
4. **Calculate the Determinant of \( A_z \)**:
   The image indicates that \( \text{det}(A_z) = -1 \).

5. **Apply Cramer's Rule for \( z \)**:
   \[
   z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{-1}{7} = -\frac{1}{7}
   \]

Thus, the solution for \( z \) is \( z = -\frac{1}{7} \).

Would you like further details on this solution or any clarification?

Here are 5 related questions to deepen understanding:
1. How is Cramer's Rule used for systems larger than \( 3 \times 3 \)?
2. What are the conditions under which Cramer's Rule can be applied?
3. Can Cramer's Rule be used if the determinant of the coefficient matrix is zero? Why or why not?
4. How do you find the determinant of a \( 3 \times 3 \) matrix manually?
5. How does Cramer's Rule compare to other methods like Gaussian elimination?

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

Use Cramer's rule and the calculator provided to find the value of z that satisfies the system of linear equations.

Learn how to solve a 3x3 system of linear equations using Cramer's Rule. The solution involves finding the determinant of the coefficient matrix and applying Cramer's formula to find the value of z. Suitable for high school students in Grades 10-12.

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