1. **Stating the problem:** We are given a 4x4 matrix \( A \) and asked to find \( z \) using Cramer's rule. The matrix \( A \) is: $$ \begin{bmatrix} 1 & 5 & 2 & 3 \\ -1 & 7 & 3 & 4 \\ 0 & 5 & 9 & 7 \\ 4 & 1 & 8 & 15 \end{bmatrix} $$ 2. **Recall Cramer's rule:** For a system \( A\mathbf{x} = \mathbf{b} \), the solution for variable \( x_i \) is: $$ x_i = \frac{\det(A_i)}{\det(A)} $$ where \( A_i \) is the matrix \( A \) with the \( i^{th} \) column replaced by \( \mathbf{b} \). 3. **Given:** \( \det(A) = 42 \) (from the text "126 / 42 = 3" implies determinant 42), and the problem states to find \( z \) using Cramer's method. 4. **Step to find \( z \):** - Replace the third column of \( A \) with the constants vector \( \mathbf{b} \). - Calculate \( \det(A_z) \). - Compute \( z = \frac{\det(A_z)}{\det(A)} \). 5. **Note:** The constants vector \( \mathbf{b} \) is not explicitly given in the problem, so we cannot compute \( \det(A_z) \) numerically here. 6. **Summary:** - Use the formula: $$ z = \frac{\det(A_z)}{42} $$ - Calculate \( \det(A_z) \) by replacing the third column of \( A \) with \( \mathbf{b} \). Since the constants vector is missing, the exact numeric value of \( z \) cannot be determined from the given data. **Final answer:** $$ z = \frac{\det(A_z)}{42} $$ where \( \det(A_z) \) is the determinant of matrix \( A \) with the third column replaced by the constants vector. This completes the solution using Cramer's rule for \( z \).