Precalculus, Chapter 8, 8.1, Section 8.1, Problem 83

The augmented matrix is
[[-1,1,-1,-14], [2,-1,1,21],[3,2,1,19]]
On applying R_1 -gt R_1 +R_2 we get (means changing 1st row as the sum of of first and second row)

[[1,0,0,7],[2,-1,1,21],[3,2,1,19]]
On applying R_2 -gt R_2 - 2R_1 we get

[[1,0,0,7], [0,-1,1,7], [3,2,1,19]]
On applying R_2 -gt -R_2 and R_3 -gt R_3 - 3R_1 we get

[[1,0,0,7],[0,1,-1,-7],[0,2,1,-2]]
On applying R_3 -gt R_3 - 2R_2 we get

[[1,0,0,7],[0,1,-1,7],[0,0,3,12]]
On applying R_3 -gt(R_3)/3 we get

[[1,0,0,7],[0,1,-1,-7],[0,0,1,4]]
Hence the given system of equations is equivalent to the following system of equations

x = 7
y - z = -7 and

z = 4
therefore the solution set is
x = 7, y =-3, z = 4