Substitute for zin the second equation, to find that Finally,
substitute 2 for yand for zin the first equation,
to determine that The solution set of the original system is
Check by substitution.
x= 1. 51 1, 2, - 126.
x-y+ 5 z=-6,
- 1
y- y=2.
83 z=
14- 1 3 ,
SECTION 4.4 Solving Systems of Linear Equations by Matrix Methods 251
NOW TRY
EXERCISE 3
Use row operations to solve
each system.
(a)
(b)
- x- 2 y=- 7
x+ 2 y= 7- 6 x+ 2 y= 4
3 x- y= 8NOW TRY ANSWERS
2.
- (a)
(b) 51 x, y 2 |x+ 2 y= 76
051 2, -1, 3 26NOW TRY
EXERCISE 2
Use row operations to solve
the system.
2 x- y+ 3 z= 14- x+ 2 y+ z=- 1
x+ y- 2 z=- 5
NOW TRYOBJECTIVE 5 Use row operations to solve special systems.
Recognizing Inconsistent Systems or Dependent EquationsUse row operations to solve each system.
(a)
Write the
augmented matrix.The corresponding system of equations is
Falsewhich has no solution and is inconsistent. The solution set is
(b)
Write the
augmented matrix.The corresponding system is
Truewhich has dependent equations. We use the second equation of the given system,
which is in standard form, to express the solution set.
51 x, y 2 | 5 x- 6 y=- 156 NOW TRY
0 = 0 ,
x-
6
5
y=- 3
- 5R 1 +R 2
c
1
0
-
6
50
`
- 3
0
d
- 101 R 1
c
1
5
-^65
- 6
`
- 3
- 15
d
c
- 10
5
12
- 6
`
30
- 15
d
5 x- 6 y=- 15
- 10 x+ 12 y= 30
0.
0 = 28 ,
x-
3
2
y= 4
6R 1 +R 2c
1
0
-
3
20
`
4
28
d
1c^12 R^1
- 6
-^32
9
`
4
4
d
c
2
- 6
- 3
9
`