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Using Cramer’s Rule to Solve a System of Linear Equations

HomeAlgebraMatrices and Determinants

Cramer’s Rule, named for the Swiss mathematician Gabriel Cramer, requires the understanding of determinants to solve a system of linear equations.

If the determinate equals zero Cramer’s Rule does not apply and the system of linear equations has either; 1) No solution, represented geometrically as parallel lines, or 2)  Has infinite solutions, where at every point the equations represent the same line.

Cramer’s Rule for Solving a 2 × 2 System:

To determine a mechanical procedure for solving a system of two linear equations in two variables observe:

a1

b1

a2

b2

Matrix A

det(A) = a1b2 – a2b1

k1

b1

k2

b2

Matrix B

det(B) = k1b2 – k2b1

Now consider two linear equations in two variables.

a1x + b1y = k1

a2x + b2y = k2

To eliminate y, multiply both sides of A by b2 and both sides of B by -b1.

a1b2x + b1b2y = k1b2

-a2b1x + -b2b1y = -k2b1

Add the two equations.

x(a1b2 − a2b1) = k1b2 − k2b1

The result is

x det(A) = det(B)

If the left-side equation determinate given by (a1b2 − a2b1) is not zero then

x = det(B) / det(A)

The determinate of the denominator has the coefficients of x and y as they are in the original equations. Call this the Determinate D:

a1x + b1y = k1

a2x + b2y = k2

=

a1

b1

a2

b2

 det(A) = D

=

The determinate of the numerator, Dx, is obtained from D by replacing the a’s (coefficients of x) by the corresponding k’s:

Dx =

k1

b1

k2

b2

So that x = Dx / D  (if D ≠ 0)

Similarly it can be shown that Dy, the 2 × 2 determinant obtained from D by replacing the b’s (coefficients of y) by the corresponding k’s is:

Dy =

a1

k1

a2

k2

And if D ≠ 0 then y = Dy / D

Thus, to solve by Cramer’s Rule two linear equations having two variables:

Linear Equations System

10x – 7y = 12

3x – 2y = 5

10

-7

3

-2

 10(-2) – 3(-7) = 1

=

D =

12

-7

5

-2

 12(-2) – 5(-7) = 11

=

Dx =

10

12

3

5

 10(5) – 3(12) = 14

=

Dy =

If D ≠ 0; then by Cramer’s Rule…

x = Dx / D = 11 / 1 = 11

y = Dy / D = 14 / 1 = 14

The unique solution to the linear equation system is (11, 14).

3×3 Cramer’s Rule
Cramer’s Rule, named for the Swiss mathematician Gabriel Cramer

Gabriel Cramer (1704-1752)

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