For the system:
\[ \begin{cases} a x + b y = c \\ d x + e y = f \end{cases} \]The determinant \( D = \det \begin{pmatrix} a & b \\ d & e \end{pmatrix} = a e - b d \).
If \( D \neq 0 \), the unique solution is:
\[ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} \]where \( D_x = \det \begin{pmatrix} c & b \\ f & e \end{pmatrix} = c e - b f \) and \( D_y = \det \begin{pmatrix} a & c \\ d & f \end{pmatrix} = a f - c d \).
If \( D = 0 \) and \( D_x = D_y = 0 \), the lines are coincident (infinite solutions).
If \( D = 0 \) and \( D_x \neq 0 \) or \( D_y \neq 0 \), the lines are parallel (no intersection).