For a system of three linear equations represented as: $\left\{\begin{array}{l} a_1 x + b_1 y + c_1 z = k_1 \\\\ a_2 x + b_2 y + c_2 z = k_2 \\\\ a_3 x + b_3 y + c_3 z = k_3 \end{array}\right.$

The solution $(x, y, z)$ can be determined using determinants: $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, and $z = \frac{D_z}{D}$

Where $D$ is the determinant of the coefficients of the variables: $D = \begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \end{vmatrix}$

$D_x$ is formed by replacing the $x$-coefficients with the constants $k_1$, $k_2$, and $k_3$: $D_x = \begin{vmatrix} k_1 & b_1 & c_1 \\\\ k_2 & b_2 & c_2 \\\\ k_3 & b_3 & c_3 \end{vmatrix}$

$D_y$ is formed by replacing the $y$-coefficients with the constants: $D_y = \begin{vmatrix} a_1 & k_1 & c_1 \\\\ a_2 & k_2 & c_2 \\\\ a_3 & k_3 & c_3 \end{vmatrix}$

$D_z$ is formed by replacing the $z$-coefficients with the constants: $D_z = \begin{vmatrix} a_1 & b_1 & k_1 \\\\ a_2 & b_2 & k_2 \\\\ a_3 & b_3 & k_3 \end{vmatrix}$