Key Concept: Cramer's Rule for Systems of Linear Equations

A system of linear equations of the form $a_1x + b_1y + c_1z = d_1$ $a_2x + b_2y + c_2z = d_2$ $a_3x + b_3y + c_3z = d_3$ can be analyzed using determinants. Let $\Delta$ be the determinant of the coefficient matrix: $\Delta = \left| {\begin{matrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{matrix} } \right|$