uses determinants to
solve systems of linear equations
. Consider the system of two linear equations in two variables:
Using the linear combination method, you can verify that
Note that the denominators are equal to the determinant of the coefficients.
The numerators are equal to the determinants and where
is formed by replacing the column of coefficients of in with the column of constants and is formed by replacing the column of coefficients of in with the column of constants.
Use determinants to solve the system of equations:
Therefore, the solution is .
Determinants can also be used to solve a system of linear equations in three variables:
This method can be generalized for a system of linear equations in variables.
It was named for the Swiss mathematician Gabriel Cramer.