Question

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

Answer 1

The solution is `((x),(y),(z))=((-1),(2),(1))`

Explanation:

Perform the Gauss-Jordan Elimination on the augmented matrix

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

Make the pivot in the first column `R1larr(R1)/2`

`((1,1/2,-3/2,|,-3/2),(3,2,4,|,5),(-4,-1,2,|,4))`

Eliminate the first column `R2larr(R2-3R1)` and `R3larr(R3+4R1)`

`((1,1/2,-3/2,|,-3/2),(0,1/2,17/2,|,19/2),(0,1,-4,|,-2))`

Make the pivot in the `2`nd column swap `R2harrR3`

`((1,1/2,-3/2,|,-3/2),(0,1,-4,|,-2),(0,1/2,17/2,|,19/2))`

Eliminate the second column `R3larr(R3-1/2(R2))`, `R1larr(R1-(R2)/2)`

`((1,0,1/2,|,-1/2),(0,1,-4,|,-2),(0,0,21/2,|,21/2))`

Make the pivot in the `3rd` column `R3larr(R3*2/21)`

`((1,0,1/2,|,-1/2),(0,1,-4,|,-2),(0,0,1,|,1))`

Eliminate the `3rd` column `R1larr(R1-(R3)/2)` and `R2larr(R2+4R3)`

`((1,0,0,|,-1),(0,1,0,|,2),(0,0,1,|,1))`

The solution is

`((x),(y),(z))=((-1),(2),(1))`