OBJECTIVE 4 Use row operations to solve a system with three equations.
Using Row Operations to Solve a System with Three VariablesUse row operations to solve the system.
Start by writing the augmented matrix of the system.
C
1
3
1
- 1
3
3
5
- 1
2
3
- 6
10
5
S
x+ 3 y+ 2 z= 5
3 x+ 3 y- z= 10
x- y+ 5 z=- 6
EXAMPLE 2
250 CHAPTER 4 Systems of Linear Equations
Write the
augmented matrix.This matrix already has 1 in row one, column one. Next get 0’s in the rest of column
one. First, add to row two the results of multiplying each number of row one by
Now add to the numbers in row three the results of multiplying each number of row
one by
Introduce 1 in row two, column two, by multiplying each number in row two by
To obtain 0 in row three, column two, add to row three the results of multiplying each
number in row two by
Obtain 1 in row three, column three, by multiplying each number in row three by
The final matrix gives this system of equations.
z=- 1
y-
8
3
z=
14
3
x-y+ 5 z=- 6
3
23 R 3C
1
0
0
- 1
1
0
5
-
8
31
3
- 6
14
3- 1
S
323.
- 4R 2 +R 3
T- 6
14
3-^233
41
0
0
- 1
1
0
5
-^83
23
3D- 4.
1C 6 R 2
1
0
0
- 1
1
4
5
-^83
- 3
3
- 6
14
311
S
16.
- 1R 1 +R 3
C
1
0
0
- 1
6
4
5
- 16
- 3
3
- 6
28
11
S
- 1.
C -3R 1 +R 2
1
0
1
- 1
6
3
5
- 16
2
3
- 6
28
5
S
- 3.
This matrix is in
row echelon form.