College Algebra 9th txtbk

SECTION 7–5 Determinants and Cramer’s Rule 493
Z THEOREM 3 Cramer’s Rule for Three Equations in Three Variables
Given the system
then
a 11 x a 12 y a 13 z k 1
a 11 a 12 a 13
a 21 x a 22 y a 23 z k 2 with D † a 21 a 22 a 23 † 0
a 31 x a 32 y a 33 z k 3 a 31 a 32 a 33
x
k 1 a 12 a 13
† k 2 a 22 a 23 †
k 3 a 32 a 33
D
y
a 11 k 1 a 13
† a 21 k 2 a 23 †
a 31 k 3 a 33
D
z
a 11 a 12 k 1
† a 21 a 22 k 2 †
a 31 a 32 k 3
D
You can easily remember these determinant formulas for x, y, and z if you observe the
following:
1. Determinant D is formed from the coefficients of x, y, and z, keeping the same relative
position in the determinant as found in the system of equations.
2. Determinant D appears in the denominators for x, y, and z.
3. The numerator for x can be obtained from D by replacing the coefficients of x (a 11 , a 21 ,
and a 31 ) with the constants k 1 , k 2 , and k 3 , respectively. Similar statements can be made
for the numerators for y and z.
EXAMPLE 5 Solving a Three-Variable System with Cramer’s Rule
Solve using Cramer’s rule:
x y 2
3y z 4
x z 3
SOLUTION
1
D † 0
1
x
y
z
2
† 4
3
1
† 0
1
1
† 0
1
1
3
0
2
4
3
2
1
3
0
2
1
3
0
2
0
1 † 2
1
0
1 †
1
7 2
0
1 †
1
3 2
2
4 †
3
1 2