2 x + 4 y + 6 z = 4
1 x + 5 y + 9 z = 2
2 x + 1 y + 3 z = 7
Solution:
can be written in augmented matrix form as:
2 | 4 | 6 | 4 |
1 | 5 | 9 | 2 |
2 | 1 | 3 | 7 |
We can then perform Gauss Elimination using the Augmented Matrix Form just as before:
make a11 = 1
<--- (row 1) / 2
2 | 4 | 6 | 4 |
1 | 5 | 9 | 2 |
2 | 1 | 3 | 7 |
make a21 = 0 & a31 = 0
1 | 2 | 3 | 2 |
1 | 5 | 9 | 2 |
2 | 1 | 3 | 7 |
<--- (row 3) - 2 (row 1)
make a22 = 0
1 | 2 | 3 | 2 |
0 | 3 | 6 | 0 |
0 | -3 | -3 |
make a32 = 0
1 | 2 | 3 | 2 |
0 | 1 | 2 | 0 |
0 | -3 | -3 |
<--- (row 3) + 3 (row 2)
make a33 = 1
1 | 2 | 3 | 2 |
0 | 1 | 2 | 0 |
0 | 0 | 3 |
<--- (row 3) / 3
now in Gauss form
1 | 2 | 3 | 2 |
0 | 1 | 2 | 0 |
0 | 0 | 1 |
This matrix, represents
1 x + 2 y + 3 z = 2
0 x + 1 y + 2 z = 0
0 x + 0 y + 1 z = 1
Gauss Form
and, as before, has solution
x = 3 , y = -2 , z = 1
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