Example 3 - Solve 3x3 Systems of Equations using Gauss Elimination

Example:
Gauss Elimination 3x3 system
 2 x + 4 y + 6 z = 4
 1 x + 5 y + 9 z = 2
 2 x + 1 y + 3 z = 7

Solution:
make a₁₁ = 1
2 x + 4 y + 6 z = 4 <---- (row 1) / 2
1 x + 5 y + 9 z = 2
2 x + 1 y + 3 z = 7

make a₂₁ = 0 & a₃₁ = 0
1 x + 2 y + 3 z = 2
1 x + 5 y + 9 z = 2 <---- (row 2) - 1 (row 1)
2 x + 1 y + 3 z = 7 <---- (row 3) - 2 (row 1)

make a₂₂ = 1
1 x + 2 y + 3 z = 2
0 x + 3 y + 6 z = 0 <---- (row 2) / 3
0 x  - 3 y  - 3 z = 3
now "zeros"... as required

make a₃₂ = 0
1 x + 2 y + 3 z = 2
0 x + 1 y + 2 z = 0
0 x   - 3 y - 3 z = 3 <---- (row 3) + 3 (row 2) or (row 3) - (-3) (row 2)

make a₃₃ = 1
1 x + 2 y + 3 z = 2
0 x + 1 y + 2 z = 0
0 x + 0 y + 3 z = 3 <---- (row 3) / 3

Now in Gauss Form
1 x + 2 y + 3 z = 2
0 x + 1 y + 2 z = 0
0 x + 0 y + 1 z = 1
The system is now in Gauss form.
(with "ones" on the diagonal and "zeroes" below the diagonal)
... from row (3): z = 1
back sub. in row (2): y = -2 (1) = -2 ;  y = -2
back sub. in row (1): x = 2 - (2) (-2) - (3) (1) = 3 ;  x = 3
\ (x , y , z) = (3 , -2 , 1)