Using Cramer’s Rule to Solve a 2 2 SystemUse Cramer’s rule to solve the system.
By Cramer’s rule, and If then we find and
From Cramer’s rule,
and
The solution set is as can be verified by checking in the given system.
NOW TRYEA
152 , -
112 BF,
y=
Dy
D
=
11
- 2
=-
11
2
x=.
Dx
D
=
- 15
- 2
=
15
2
Dy= `
5
6
- 1
1
` = 5112 - 1 - 126 = 11
Dx= `
- 1
1
7
8
(^) ` =- 1182 - 7112 = - 15
D= `
5
6
7
8
(^) ` = 5182 - 7162 = - 2
DZ0, Dx Dy.
Dy
Dy=.
Dx
Dx=
6 x+ 8 y= 1
5 x+ 7 y=- 1
EXAMPLE 4 :
APPENDIX A Determinants and Cramer’s Rule 719
Cramer’s Rule for 3 3 SystemsFor the system
with
the values of x, y, and zare given by
and z
Dz
D
y.
Dy
D
x ,
Dx
D
,
D=^3
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
D^3 Z0,
z=
3
a 1
a 2
a 3
b 1
b 2
b 3
d 1
d 2
d 3
(^3) ,
Dy=^3
a 1
a 2
a 3
d 1
d 2
d 3
c 1
c 2
c 3
D^3 ,
x=
3
d 1
d 2
d 3
b 1
b 2
b 3
c 1
c 2
c 3
(^3) ,
a 3 x+ b 3 y+ c 3 z=d 3
a 2 x+ b 2 y+ c 2 z=d 2
a 1 x+b 1 y+ c 1 z =d 1
:
Using Cramer’s Rule to Solve a 3 3 SystemUse Cramer’s rule to solve the system.
x- 2 y+ 3 z= 4
2 x- y+ z=- 5
x+ y- z=- 2
EXAMPLE 5 :
NOW TRY
EXERCISE 4
Use Cramer’s rule to solve
the system.
2 x+ 3 y=- 93 x- 2 y=- 33NOW TRY ANSWER
- 51 - 9, 3 26