`2x-y+3z=24`

`2y-z=14`

`7x-5y=6`

Write the equations as,

`[[2,-1,3,|24],[0,2,-1,|14],[7,-5,0,|6]]`

Make the pivot in the first column by dividing the First row by 2,

`[[1,-1/2,3/2,|12],[0,2,-1,|14],[7,-5,0,|6]]`

Multiply the first row by 7,

`[[7,-7/2,21/2,|84],[0,2,-1,|14],[7,-5,0,|6]]`

Subtract the 1st row from the 3rd row and restore it,

`[[1,-1/2,3/2,|12],[0,2,-1,|14],[0,-3/2,-21/2,|-78]]`

Make the pivot in the second column by...

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`2x-y+3z=24`

`2y-z=14`

`7x-5y=6`

Write the equations as,

`[[2,-1,3,|24],[0,2,-1,|14],[7,-5,0,|6]]`

Make the pivot in the first column by dividing the First row by 2,

`[[1,-1/2,3/2,|12],[0,2,-1,|14],[7,-5,0,|6]]`

Multiply the first row by 7,

`[[7,-7/2,21/2,|84],[0,2,-1,|14],[7,-5,0,|6]]`

Subtract the 1st row from the 3rd row and restore it,

`[[1,-1/2,3/2,|12],[0,2,-1,|14],[0,-3/2,-21/2,|-78]]`

Make the pivot in the second column by dividing the 2nd row by 2,

`[[1,-1/2,3/2,|12],[0,1,-1/2,|7],[0,-3/2,-21/2,|-78]]`

Multiply the 2nd row by -1/2,

`[[1,-1/2,3/2,|12],[0,-1/2,1/4,|-7/2],[0,-3/2,-21/2,|-78]]`

subtract the 2nd row from the 1st row and restore it,

`[[1,0,5/4,|31/2],[0,1,-1/2,|7],[0,-3/2,-21/2,|-78]]`

Multiply the 2nd row by -3/2

`[[1,0,5/4,|31/2],[0,-3/2,3/4,|-21/2],[0,-3/2,-21/2,|-78]]`

subtract the 2nd row from the 3rd row and restore it,

`[[1,0,5/4,|31/2],[0,1,-1/2,|7],[0,0,-45/4,|-135/2]]`

Make the pivot in the 3rd column by dividing the 3rd row by -45/4,

`[[1,0,5/4,|31/2],[0,1,-1/2,|7],[0,0,1,|6]]`

Multiply the 3rd row by 5/4,

`[[1,0,5/4,|31/2],[0,1,-1/2,|7],[0,0,5/4,|15/2]]`

subtract the 3rd row from the first row and restore it,

`[[1,0,0,|8],[0,1,-1/2,|7],[0,0,1,|6]]`

Multiply the 3rd row by -1/2,

`[[1,0,0,|8],[0,1,-1/2,|7],[0,0,-1/2,|-3]]`

subtract the 3rd row from the 2nd row and restore it,

`[[1,0,0,|8],[0,1,0,|10],[0,0,1,|6]]`

So the solutions are x=8,y=10,z=6