Precalculus, Chapter 8, 8.1, Section 8.1, Problem 82

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