Cramer's Rule — Solving Linear Equations

Cramer's Rule uses determinants to solve a system of linear equations directly, without step-by-step elimination. Named after the Swiss mathematician Gabriel Cramer, it is among the most examined applications of determinants in commerce mathematics, because so many business problems — splitting a budget between two products, meeting two or three simultaneous constraints — end up as a small linear system.

Two equations in two unknowns. For $a_1x + b_1y = c_1, \qquad a_2x + b_2y = c_2,$ define three determinants: $D=\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix},\quad D_x=\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix},\quad D_y=\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}.$ Here $D$ is the determinant of the coefficients exactly as written; $D_x$ is $D$ with the $x$-column replaced by the constants $(c_1,c_2)$; $D_y$ is $D$ with the $y$-column replaced the same way. Provided $D \neq 0$, the unique solution is $x = \frac{D_x}{D}, \qquad y = \frac{D_y}{D}.$

Three equations in three unknowns. The rule extends unchanged to $a_1x+b_1y+c_1z=d_1,\quad a_2x+b_2y+c_2z=d_2,\quad a_3x+b_3y+c_3z=d_3.$ Form $D$ from the coefficient matrix, and form $D_x, D_y, D_z$ by replacing the $x$-, $y$-, $z$-columns respectively with the constants $(d_1,d_2,d_3)$. Then, provided $D\neq 0$, $x=\frac{D_x}{D},\qquad y=\frac{D_y}{D},\qquad z=\frac{D_z}{D}.$ …