College Algebra 9th txtbk

492 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
Z THEOREM 2 Cramer’s Rule for Two Equations and Two Variables
Given the system
then
a 11 x a 12 y k 1
a 11 a 12
with D ` ` 0
a 21 x a 22 y k 2 a 21 a 22
x
k 1 a 12
` `
k 2 a 22
D
and
y
a 11 k 1
` `
a 21 k 2
D
The determinant D is called the coefficient determinant. If D 0, then the system has
exactly one solution, which is given by Cramer’s rule. If, on the other hand, D 0, then it
can be shown that the system is either inconsistent and has no solutions or is dependent
and has an infinite number of solutions. In that case, we would need to use other methods
to determine the exact nature of the solutions.
EXAMPLE 4 Solving a Two-Variable System with Cramer’s Rule
Solve using Cramer’s rule: 3x 5y 2
4x 3y 1
SOLUTIONS
First find the determinant of the coefficient matrix:
3 5
D ` 9 20 11
4 3`
Now replace the x column with the constants and find the determinant, then divide by 11.
Now repeat, this time replacing the y column with the constants.
y
x
2 5
`
1 3 `
11
3 2
`
4 1 ` 3 (8)
11 11
The solution to the system is x 1 .
11 , y 5 11
6 5
11 1 11
5
11
MATCHED PROBLEM 4
Solve using Cramer’s rule:
3x 2y 4
4x 3y 10
Cramer’s rule can be generalized completely for any size linear system that has the
same number of variables as equations. However, it cannot be used to solve systems where
the number of variables is not equal to the number of equations. In Theorem 3 we state without
proof Cramer’s rule for three equations in three variables.