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.
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
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
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
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: February 2015
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