http://www.ck12.org Chapter 8. Systems and Matrices
w+x+z= 11
w+x= 9
x+y= 7
y+z= 5Solution: While substitution would work in this problem, the idea is to demonstrate how augmented matrices will
work even with larger matrices.
1 1 0 1
1 1 0 0
0 1 1 0
0 0 1 1
∣∣
∣∣
∣∣
∣∣
11
9
7
5
→
→ IV
→ II →
→ III →
1 1 0 1
0 1 1 0
0 0 1 1
1 1 0 0
∣∣
∣∣
∣∣
∣∣
11
7
5
9
→
→
→
→ −I →
1 1 0 1
0 1 1 0
0 0 1 1
0 0 0 − 1
∣∣
∣∣
∣∣
∣∣
11
7
5
− 2
→
→
→
→ ·(− 1 ) →
1 1 0 1
0 1 1 0
0 0 1 1
0 0 0 1
∣∣
∣∣
∣∣
∣∣
11
7
5
2
→ −IV →
→
→ −IV →
→
1 1 0 0
0 1 1 0
0 0 1 0
0 0 0 1
∣∣
∣∣
∣∣
∣∣
9
7
3
2
→
→ −III →
→
→
1 1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
∣∣
∣∣
∣∣
∣∣
9
4
3
2
→ −II →
→
→
→
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
∣∣
∣∣
∣∣
∣∣
5
4
3
2
w= 5 ,x= 4 ,y= 3 ,z= 2
Concept Problem Revisited
If you were to write the system as a matrix system you could write:
5 x+y= 6
[ x+y=^10
5 1
1 1]
·
[x
y]
=
[ 6
10
]
Vocabulary
Anaugmented matrixis a matrix formed when two matrices are joined together and operated on as if they were a
single matrix.
Guided Practice
- Use an augmented matrix to solve the following system.
2 x+y+z= 16
2 y+ 6 z= 0
x+y= 10- Use an augmented matrix to solve the following system.