Example 4 - Solve 3x3 Systems of Equations using Gauss Elimination

Example:
Gauss - Elimination 3x3 system
2 x  - 3 y + 3 z = 7
3 x + 1 y - 2 z = -11
5 x  - 2 y + 4 z = 11

Solution:
nasty coefficients
make a₁₁ = 1
2 x - 3 y + 3 z = 7 <---- (row 1) / 2 or R1 / 2
3 x + 1 y - 2 z = -11
5 x - 2 y + 4 z = 11

make a₂₁ = 0 & a₃₁ = 0
1 x   - 1.5 y + 1.5 z = 3.5
3 x  + 1    y - 2     z = -11 <---- (row 2) - 3 (row 1)
5 x   - 2    y + 4     z = 11 <---- (row 3) - 5 (row 1) or R3 - 5 R1

make a₂₂ = 1
1 x  - 1.5 y + 1.5 z = 3.5
0 x + 5.5 y  - 6.5 z = -21.5 <---- (row 2) / 5.5
0 x + 5.5 y  - 3.5 z = -6.5

make a₃₂ = 0
1 x - 1.5 y + 1.5  z = 3.5
0 x + 1   y - 1.18 z = -3.91
0 x + 5.5 y - 3.5 z = -6.5 <---- (row 3) - 5.5 (row 2)

make a₃₃ = 1
1 x - 1.5 y + 1.5 z = 3.5
0 x + 1 y - 1.18 z = -3.91
0 x + 0 y  + 3    z = 15 <---- (row 3) / 3

Now in Gauss Form
1 x - 1.5 y + 1.5 z = 3.5
0 x + 1 y - 1.18 z = -3.91
0 x + 0 y + 1    z = 5
"ones" on the diagonal and "zeroes" below the diagonal

... from row (3): z = 5
back sub. in row (2) --> y = -3.91 + 1.18 (5) = 2.0 ; y = 2.0
back sub. in row (1) --> x = (1.5) (2.0) - (1.5) (5) + 3.5 = -1 ; x = -1
\ answer (x , y , z) = (-1 , 2 , 5)