CK-12-Pre-Calculus Concepts

8.8. Cramer’s Rule http://www.ck12.org

8.8 Cramer’s Rule

Here you will solve systems of equations using Cramer’s Rule.
A system of equations can be represented and solved in general using matrices and determinants. This method can
be significantly more efficient than eliminating variables in equations. What does it mean for a solution method to
be more efficient? Is Cramer’s Rule the most efficient means of solving a system of equations?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/61652

http://www.youtube.com/watch?v=ItxF3IjC_uw James Sousa: Cramer’s Rule

Guidance

The determinant is defined in a seemingly arbitrary way; however, when you look at the general solution for a
2 ×2 matrix, the reasoning why it is defined this way is apparent.

ax+by=e

cx+dy=f

When you solve the system above foryandx, you get the following:

y=a fad−−cebc

x=b fad−−debc

Note that the system can be represented by the matrix and the solutions can be written as ratios of two determi-
nants. The determinant in the denominator is of the coefficient matrix. The numerator of thexsolution is the
determinant of the new matrix whose columns are made up of theycoefficients and the solution coefficients. The
numerator of theysolution is the determinant of the new matrix made up of thexcoefficients and the solution
coefficients.