Cramer’s Rule uses determinants to solve systems of linear equations. Consider the system of two linear equations in two variables:
ax + by = c
dx + ey = f
Using the linear combination method, you can verify that
if ae – bd ≠ 0
Note that the denominators are equal to the determinant of the coefficients.
The numerators are equal to the determinants Dx and Dy where
Dx is formed by replacing the column of coefficients of x in D with the column of constants and Dyis formed by replacing the column of coefficients of y in D with the column of constants.
Use determinants to solve the system of equations:
Therefore the solution is (2, 0).
Determinants can also be used to solve a system of linear equations in three variables:
This method can be generalized for a system of n linear equations in n variables.
It was named for the Swiss mathematician Gabriel Cramer.