structural:summarization, information:classification
Geometric interpretation

Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns presented here.

Given the system of equations

it can be considered as an equation between vectors

The area of the parallelogram determined by (a 11 a 21) {\displaystyle {\binom {a_{11}}{a_{21}}}} and (a 12 a 22) {\displaystyle {\binom {a_{12}}{a_{22}}}} is given by the determinant of the system of equations:

In general, when there are more variables and equations, the determinant of n vectors of length n will give the volume of the parallelepiped determined by those vectors in the n -th dimensional Euclidean space.

Therefore, the area of the parallelogram determined by x 1 (a 11 a 21) {\displaystyle x_{1}{\binom {a_{11}}{a_{21}}}} and (a 12 a 22) {\displaystyle {\binom {a_{12}}{a_{22}}}} has to be x 1 {\displaystyle x_{1}} times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by (b 1 b 2) = x 1 (a 11 a 21) + x 2 (a 12 a 22) {\displaystyle {\binom {b_{1}}{b_{2}}}=x_{1}{\binom {a_{11}}{a_{21}}}+x_{2}{\binom {a_{12}}{a_{22}}}} and (a 12 a 22). {\displaystyle {\binom {a_{12}}{a_{22}}}.}

Equating the areas of this last and the second parallelogram gives the equation

from which Cramer's rule follows.