Cramer's Rule
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Cramer's Rule is a method for solving systems of linear equations with the same number of equations as variables. It is named after Gabriel Cramer, a Swiss mathematician who first published the rule in 1750.
Cramer's Rule Formula
Cramer's Rule states that the solution to a system of linear equations is given by,
Solution of x = (Determinant of numerator)/Determinant of whole matrix
where the determinant of the numerator is the determinant of the matrix formed by replacing the column of coefficients of the variable being solved for with the column of constants, and the determinant of the whole matrix is the determinant of the matrix of coefficients.
Example
For example, consider the following system of equations:
x + 2y = 5
3x + 4y = 8
Using Cramer's Rule, we can solve for x as follows:
Determinant of numerator =
|5 2|
|8 4| = 20 - 16 = 4
Determinant of whole matrix =
|1 2|
|3 4| = 4 - 6 = -2
So, x = 4/-2 = -2.
Similarly, we can solve for y as follows:
Determinant of numerator =
|1 5|
|3 8| = 8 - 15 = -7
Determinant of whole matrix =
|1 2|
|3 4| = 4 - 6 = -2
So, y = -7/-2 = 3.5.
