Cramer's Rule

Cramer's rule is a method of solving simultaneous equations for unknowns.

• Any system of equations of this kind can be written in the form
 (1)

• The coefficients on the left side of Eq.'s 1 can be written as a matrix,
 (2)

• The value of any of the can be found via
 (3)

where the notation and denote the determinants of matrices and , and the matrix is obtained by replacing the column of matrix with the coefficients on the left side of Eq.'s 1. For example,
 (4)

### The Case

• Two equations involving two unknowns can be written in the form
 (5)

which yields
 (6)

• Eq. 3 gives
 (7)

and
 (8)

### An Example

• The system of equations
 (9)

has solutions
 (10)

and
 (11)

• The results can be verified by substituting them into either of Eq.'s 9,
 (12)

### An Example

• The system of equations
 (13)

has solutions
 (14)

 (15)

and
 (16)

• The results can be verified by substituting them into any of Eq.'s 13,
 (17)

 Copyright © 2002-2004, Lewis A. Riley Updated Mon Jan 19 13:29:10 2004