Cramer's Rule — Solving Systems of Linear Equations

Cramer's Rule uses determinants to solve a system of linear equations directly, without the row-reduction (elimination) method usually taught alongside it. It is named after the Swiss mathematician Gabriel Cramer, and it is one of the most tested applications of determinants in Business Mathematics, because so many business problems — allocating a fixed budget across two products, balancing two or three business constraints simultaneously — end up as a small system of linear equations.

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

Three equations in three unknowns. The rule extends exactly the same way 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 $D_x, D_y, D_z$ by replacing the $x$-, $y$-, and $z$-coefficient columns respectively with the constants column $(d_1,d_2,d_3)$. Then, provided $D\neq 0$, $x=\dfrac{D_x}{D},\qquad y=\dfrac{D_y}{D},\qquad z=\dfrac{D_z}{D}.$ …