\[ \begin{array}{r}3 x+7 y+z=10 \\ -3 x-4 y-z=6 \\ -5 x-6 y-2 z=8\end{array} \] Write the four relevant determinants in any order:

To find the relevant determinants for the given system of equations, we first need to form the coefficient matrix and the associated determinants. The equations are: 1. \(3x + 7y + z = 10\) 2. \(-3x - 4y - z = 6\) 3. \(-5x - 6y - 2z = 8\) The coefficient matrix \(A\) is: \[ A = \begin{bmatrix} 3 & 7 & 1 \\ -3 & -4 & -1 \\ -5 & -6 & -2 \end{bmatrix} \] The determinants we will find are: 1. The determinant of the coefficient matrix \(A\), denoted as \(|A|\). 2. The determinant of matrix \(A_x\) which is formed by replacing the first column of \(A\) with the constants from the right side of the equations. 3. The determinant of matrix \(A_y\) which is formed by replacing the second column of \(A\) with the constants from the right side. 4. The determinant of matrix \(A_z\) which is formed by replacing the third column of \(A\) with the constants from the right side. The constants from the right side are \([10, 6, 8]^T\). Thus, the matrices are: \[ A_x = \begin{bmatrix} 10 & 7 & 1 \\ 6 & -4 & -1 \\ 8 & -6 & -2 \end{bmatrix}, \quad A_y = \begin{bmatrix} 3 & 10 & 1 \\ -3 & 6 & -1 \\ -5 & 8 & -2 \end{bmatrix}, \quad A_z = \begin{bmatrix} 3 & 7 & 10 \\ -3 & -4 & 6 \\ -5 & -6 & 8 \end{bmatrix} \] So the four relevant determinants you may calculate are \(|A|\), \(|A_x|\), \(|A_y|\), and \(|A_z|\).