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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)
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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
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: February 2015
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