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Monday, June 13, 2011

Example 5 - Solve 3x3 Systems of Equations using Gauss Elimination (Augmented Matrix)

Example:
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 2) - 1 (row 1)
<--- (row 3) - 2 (row 1)


make a22 = 0
1
 2
3
2
0
 3
6
0
0
-3
-3
3
<--- (row 2) / 3


make a32 = 0
1
2
3
2
0
1
2
0
0
-3
-3
3

<--- (row 3) + 3 (row 2)

make a33 = 1
1
2
3
2
0
1
2
0
0
0
3
3

<--- (row 3) / 3

now in Gauss form
1
2
3
2
0
1
2
0
0
0
1
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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