Question

How do you solve using gaussian elimination or gauss-jordan elimination, `10x-7y+3z+5u=6`, `-6x+8y-z-4u=5`, `3x+y+4z+11u=2`, `5x-9y-2z+4u=7`?

Answer 1

`x=(22+7xx(44/19))/10=3.8210526315789473684210526315789`

`y=44/19=2.3157894736842105263157894736842`

`z=-7`

`u=1`

Explanation:

`([10,-7,3,5,|,6],[-6,8,-1,-4,|,5],[3,1,4,11,|,2],[5,-9,-2,4,|,7])~~([10,-7,3,5,|,6],[-6,8,-1,-4,|,5],[6,2,8,22,|,4],[10,-18,-4,8,|,14])`
`R_4=R_4xx2`
`R_3=R_3xx2`
`R_4=R_4-R_1`
`R_3=R_3+R_2`

`([10,-7,3,5,|,6],[-6,8,-1,-4,|,5],[0,10,7,18,|,9],[0,-11,-7,3,|,8])~~([30,-21,9,15,|,18],[-30,40,-5,-20,|,25],[0,10,7,18,|,9],[0,-11,-7,3,|,8])`
`R_1=R_1xx3`
`R_2=R_2xx5`
`R_2=R_2+R_1`

`([30,-21,9,15,|,18],[0,19,4,-5,|,43],[0,10,7,18,|,9],[0,-11,-7,3,|,8])~~([30,-21,9,15,|,18],[0,19,4,-5,|,43],[0,10,7,18,|,9],[0,-110,-70,30,|,80])`
`R_4=R_4*10`
`R_4=R_4+11xxR_3`

`([30,-21,9,15,|,18],[0,19,4,-5,|,43],[0,10,7,18,|,9],[0,0,7,228,|,179])~~([30,-21,9,15,|,18],[0,190,40,-50,|,430],[0,190,133,342,|,171],[0,0,7,228,|,179])`
`R_3=R_3xx19`
`R_2=R_2xx10`
`R_3=R_3-R_2`

`([30,-21,9,15,|,18],[0,190,40,-50,|,430],[0,0,93,392,|,-259],[0,0,7,228,|,179])~~([30,-21,9,15,|,18],[0,190,40,-50,|,430],[0,0,651,2744,|,-1813],[0,0,651,21204,|,16647])`
`R_4=R_4xx93`
`R_3=R_3xx7`
`R_4=R_4-R_3`

`([30,-21,9,15,|,18],[0,190,40,-50,|,430],[0,0,651,2744,|,-1813],[0,0,0,18460,|,18460])~~([10,-7,3,5,|,6],[0,19,4,-5,|,43],[0,0,93,392,|,-259],[0,0,0,1,|,1])`
Dividing what is possible
`R_3=R_3-392xxR_4`
`R_2=R_2+5xxR_4`
`R_2=R_2-5xxR_4`

`([10,-7,3,0,|,1],[0,19,4,0,|,48],[0,0,93,0,|,-651],[0,0,0,1,|,1])~~([10,-7,0,0,|,22],[0,19,0,0,|,44],[0,0,1,0,|,-7],[0,0,0,1,|,1])`
`R_3=R_3xx1/93`
`R_2=R_2-4xxR_3`
`R_1=R_1-3xxR_3`
`R_2=R_2xx1/19`
`R_1=R_1+7xxR_2`

`([1,0,0,0,|,(22+7xx(44/19))/10],[0,1,0,0,|,44/19],[0,0,1,0,|,-7],[0,0,0,1,|,1])`

`x=(22+7xx(44/19))/10=3.8210526315789473684210526315789`

`y=44/19=2.3157894736842105263157894736842`

`z=-7`

`u=1`