Question

How do you solve using gaussian elimination or gauss-jordan elimination, `‎2x+y-z+2w=-6`, `3x+4y+w=1`, `x+5y+2z+6w=-3`, `5x+2y-z-w=3`?

Answer 1

`P="[(-7, 0, -4, 2)]"`

Explanation:

`([2,1,-1,2,|,-6],[3,4,0,1,|,1],[1,5,2,6,|,-3],[5,2,-1,-1,|,3])~~([1,5,2,6,|,-3],[3,4,0,1,|,1],[2,1,-1,2,|,-6],[5,2,-1,-1,|,3])`
`R_1hArrR_3`

---

`R_2=R_2-3xxR_1`
`R_3=R_3-2xxR_1`
`R_4=R_4-5xxR_1`

`([1,5,2,6,|,-3],[0,-11,-6,-17,|,10],[0,-9,-5,-10,|,0],[0,-23,-11,-31,|,18])`
`R_2=R_2xx-9`
`R_3=R_3xx11`

`([1,5,2,6,|,-3],[0,99,54,153,|,-90],[0,-99,-55,-110,|,0],[0,-23,-11,-31,|,18])`
`R_2=R_2+R_3`
then `R_2hArrR_3`

`([1,5,2,6,|,-3],[0,-99,-55,-110,|,0],[0,0,-1,43,|,-90],[0,-23,-11,-31,|,18])`
`R_2=R_2xx23/11`
`R_4=R_4xx9`

`([1,5,2,6,|,-3],[0,-207,-115,-230,|,0],[0,0,-1,43,|,-90],[0,-207,-99,-279,|,162])`
`R_4=R_4-R_2`

`([1,5,2,6,|,-3],[0,-207,-115,-230,|,0],[0,0,-1,43,|,-90],[0,0,16,-49,|,162])`
`R_2=R_2xx1/23`
`R_3=R_3xx16`

`([1,5,2,6,|,-3],[0,-9,-5,-10,|,0],[0,0,-16,688,|,-1440],[0,0,16,-49,|,162])`
`R_4=R_4+R_3`

`([1,5,2,6,|,-3],[0,-9,-5,-10,|,0],[0,0,-16,688,|,-1440],[0,0,0,-639,|,-1278])`
`R_4=R_4xx-1/639`
`R_3=R_3xx-1/16`

`([1,5,2,6,|,-3],[0,-9,-5,-10,|,0],[0,0,1,-43,|,-90],[0,0,0,1,|,2])`
`R_3=R_3+43xxR_4`
`R_2=R_2+10xxR_4`
`R_1=R_1-6xxR_4`

`([1,5,2,0,|,-15],[0,-9,-5,0,|,20],[0,0,1,0,|,-4],[0,0,0,1,|,2])`
`R_2=R_2+5xxR_3`
`R_1=R_1-2xxR_3`

`([1,5,0,0,|,-7],[0,-9,0,0,|,0],[0,0,1,0,|,-4],[0,0,0,1,|,2])`
`R_2=R_2xx-1/9`
`R_1=R_1-5xxR_2`

`([1,0,0,0,|,-7],[0,1,0,0,|,0],[0,0,1,0,|,-4],[0,0,0,1,|,2])`

`P="[(-7, 0, -4, 2)]"`