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Gabriel Cramer (1704-1752)

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.

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 |

D =

10(-2) – 3(-7) = 1

=

12(-2) – 5(-7) = 11

=

12 | -7 |

5 | -2 |

Dx =

10 | 12 |

3 | 5 |

Dy =

10(5) – 3(12) = 14

=

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).