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Cramer’s Rule

Cramer’s Rule uses determinants to solve systems of linear equations .  Consider the system of two linear equations in two variables:
a x + b y = c d x + e y = f

Using the linear combination method, you can verify that

x = c e b f a e b d and y = a f c d a e b d if a e b d 0

Note that the denominators are equal to the determinant of the coefficients.

D = | a b d e |

The numerators are equal to the determinants D x and D y where

D x = | c b f e | and D y = | a c d f |

D x is formed by replacing the column of coefficients of x in D with the column of constants and D y is formed by replacing the column of coefficients of y in D with the column of constants.

By substitution,

x = | c b f e | | a b d e | = D x D and y = | a c d f | | a b d e | = D y D .

Example:

Use determinants to solve the system of equations: { x 5 y = 2 2 x + y = 4

x = D x D = | 2 5 4 1 | | 1 5 2 1 | = 2 + 20 1 + 10 = 22 11 = 2 y = D y D = | 1 2 2 4 | | 1 5 2 1 | = 4 4 1 + 10 = 0 11 = 0

Therefore, the solution is ( 2 , 0 ) .

Determinants can also be used to solve a system of linear equations in three variables:

a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3

Then,

D = | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | 0 D x = | d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 | D y = | a 1 d 1 c 1 a 2 d 2 c 2 a 3 d 3 c 3 | D z = | a 1 b 1 d 1 a 2 b 2 d 2 a 3 b 3 d 3 |

And,

x = D x D , y = D y D , z = D z D .

This method can be generalized for a system of n linear equations in n variables.

It was named for the Swiss mathematician Gabriel Cramer.