Cramer's Rule to Solve Systems of Equations

This is a tutorial on proving cramer's rule of solving 2 by 2 systems of linear equations. Rules for 3 by 3 systems of equations are also presented. To check answers when solving 2 by 2 and 3 by 3 systems of equations, you may want to use these online Systems of Equations Calculator and Solver. .



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To find rules (or formulas) that may be used solve any system of equations, we need to solve the general system of the form

a 1 x + b 1 y = c 1    (1)

a 2 x + b 2 y = c 2    (2)


We multiply equation (1) by b 2 and equation (2) by -b 1 and add the right and left hand terms.

b 2 a 1 x + b 2 b 1 y = b 2 c 1    (1)

- b 1 a 2 x - b 1 b 2 y = - b 1 c 2    (2)
____________________________

b 2 a 1 x - b 1 a 2 x = b 2 c 1 - b 1 c 2

Assuming that a 1 b 2 - a 2 b 1 is not equal to zero, solve the above equation for x to obtain.

x = ( c 1 b 2 - c 2 b 1 ) / ( a 1 b 2 - a 2 b 1 )

We can use similar steps to eliminate x and solve for y to obtain.

y = ( a 1 c 2 - a 2 c 1 ) / ( a 1 b 2 - a 2 b 1 )


Using the determinant of a Matrix notation, the solution to the given 2 by 2 system of linear equations is given by

x = D x / D and y = D y / D

where D, D x and D y are the determinants defined by

cramer's rule for 2 by 2 system of linear equations.


For a 3 by 3 system of linear equations of the form

a 1 x + b 1 y + c 1 z = d 1    (1)

a 2 x + b 2 y + c 2 z = d 2    (2)

a 3 x + b 3 y + c 3 z = d 3    (3)


Cramer's rule gives the solution as follows

x = D x / D , y = D y / D and z = D z / D

where D, D x, D y and D z are determinants defined by

cramer's rule for 3 by 3 system of linear equations.


Example: Use Cramer's rule for a 3 by 3 system of linear equations to solve the following system

2 x - y + 3 z = - 3

- x - y + 3 z = - 6

x - 2y - z = - 2

Solution:

Determinants D, D x, D y and D z are given by (see how to Calculate Determinant of a Matrix.)

D = 21

D x = 21

D y = 42

D z = -21

We now use cramer's rule to find the solution of the system of equations

x = D x / D= 1

y = D y / D= 2

z = D z / D= -1

As an exercise check that (1,2,-1) is a solution to the given system of equations.

References and links related to systems of equations and determinant.

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Updated: 26 November 2007 (A Dendane)