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
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.
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Updated: 2 April 2013
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