Question

How do you solve using gaussian elimination or gauss-jordan elimination, `2x–3y+2z=2`, `x+4y-z=9`, `-3x+y–5z=5`?

Answer 1

`x=115/31`, `y=32/31` and `z=-36/31`

Explanation:

Perform the Gauss Jordan elimination on the augmented matrix

`A=((2,-3,2,|,2),(1,4,-1,|,9),(-3,1,-5,|,5))`

I have written the equations not in the sequence as in the question in order to get `1` as pivot.

Perform the folowing operations on the rows of the matrix

`R1larrR1-2R2`; `R3larrR3+3R2`

`A=((0,-11,4,|,-16),(1,4,-1,|,9),(0,13,-16,|,32))`

`R3larr(R3+R1)`

`A=((0,-11,4,|,-16),(1,4,-1,|,9),(0,2,-12,|,16))`

`R3larr(R3)/2`

`A=((0,-11,4,|,-16),(1,4,-1,|,9),(0,1,-6,|,8))`

`R1larrR1+11R3; R2larrR2-4R3`

`A=((0,0,-62,|,72),(1,0,23,|,-23),(0,1,-6,|,8))`

`R1larr(R1)/(-62)`

`A=((0,0,1,|,-36/31),(1,0,23,|,-23),(0,1,-6,|,8))`

`R2larrR2-23R1; R3larrR3+6R1`

`A=((0,0,1,|,-36/31),(1,0,0,|,115/31),(0,1,0,|,32/31))`

Thus, `x=115/31`, `y=32/31` and `z=-36/31`