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

To find the relevant determinants for the given system of equations, we need to compute the following: 1. The main determinant \( D \) formed by the coefficients of \( x, y, z \). 2. The determinant \( D_x \) formed by substituting the constant terms into the first column. 3. The determinant \( D_y \) formed by substituting the constant terms into the second column. 4. The determinant \( D_z \) formed by substituting the constant terms into the third column. Let’s compute them step-by-step: 1. The main determinant \( D \): \[ D = \begin{vmatrix} 3 & 7 & 1 \\ -3 & -4 & -1 \\ -5 & -6 & -2 \end{vmatrix} \] 2. The determinant \( D_x \): \[ D_x = \begin{vmatrix} 10 & 7 & 1 \\ 6 & -4 & -1 \\ 8 & -6 & -2 \end{vmatrix} \] 3. The determinant \( D_y \): \[ D_y = \begin{vmatrix} 3 & 10 & 1 \\ -3 & 6 & -1 \\ -5 & 8 & -2 \end{vmatrix} \] 4. The determinant \( D_z \): \[ D_z = \begin{vmatrix} 3 & 7 & 10 \\ -3 & -4 & 6 \\ -5 & -6 & 8 \end{vmatrix} \] So, the four relevant determinants are: 1. \( D \) 2. \( D_x \) 3. \( D_y \) 4. \( D_z \)