*n*linear equations in

*n*variables whose determinant is not equal to zero.

As a general rule, given the equations \(A_{1}\)

*x*+ \(B_{1}\)

*y*= \(C_{1}\) and \(A_{2}\)

*x*+ \(B_{2}\)

*y*= \(C_{2}\), then :

D = \(\begin{pmatrix}A_{1} & B_{1}\\A_{2} & B_{2}\end{pmatrix}\) = *A*\(_{1}\)× *B*\(_{2}\) – *A*\(_{2}\)× *B*\(_{1}\)

D\(_{1}\) = \(\begin{pmatrix}C_{1} & B_{1}\\C_{2} & B_{2}\end{pmatrix}\) = *C*\(_{1}\)× *B*\(_{2}\) – *B*\(_{1}\)× *C*\(_{2}\)

D\(_{2}\) = \(\begin{pmatrix}A_{1} & C_{1}\\A_{2} & C_{2}\end{pmatrix}\) = *A*\(_{1}\)× *C*\(_{2}\) – *A*\(_{2}\)× *C*\(_{1}\)

*x* = \(\dfrac{D_{1}}{D}\) where D ≠ 0 and *y* = \(\dfrac{D_{2}}{D}\) where D ≠ 0

### Example

Given the equations 3*x* + 2*y* = 12 and 5*x* + 2*y* = 16, then :

D = \(\begin{pmatrix}3 & 2\\5 & 2\end{pmatrix}\) = 3 × 2 – 5 × 2 = –4

D\(_{1}\) = \(\begin{pmatrix}12 & 2\\16 & 2\end{pmatrix}\) = 12 × 2 – 2 × 16 = –8

D\(_{2}\) = \(\begin{pmatrix}3 & 12\\5 & 16\end{pmatrix}\) = 3 × 16 – 5 × 12 = –12

*x* = \(\dfrac{–8}{–4}\) = 2 and *y* = \(\dfrac{–12}{–4}\) = 3

### Historical note*

Gabriel Cramer (1704-1752) was a Swiss mathematician who, after attempting to solve a system of five equations in five unknowns using the general elimination method, wrote in his work *Introduction à l’analyse des lignes courbes algébriques* (Introduction to the Analysis of Algebraic Curves) : “*I believe I may have found a convenient and general rule for any given number of equations and unknowns that do not have a degree higher than one* … ” [Translation] Although Cramer came to be known mainly for this rule, credit rightly belongs to Colin Maclaurin (1698-1746), a Scottish mathematician who described the rule in his *Treatise of Algebra*, which was published in 1748, two years after his death.

*Source: Jean-Paul Collette, *Histoire des mathématiques*, Volume 2, pp. 101-104.