It looks like you're talking about row-reducing an augmented matrix to solve the system of equations. Your answer is almost correct. The last row should read 0, 0, 1, 2/7.
The given system translates to
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 1 & -1 & 2 & 2 \\ 1 & 2 & -3 & 4 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%201%20%26%20-1%20%26%202%20%26%202%20%5C%5C%201%20%26%202%20%26%20-3%20%26%204%20%5Cend%7Barray%7D%20%5Cright%5D)
Eliminate x from the last two rows by combining -2 (row 2) and row 1, and -2 (row 3) and row 1; that is,
(2x - 3y + z) - 2 (x - y + 2z) = 2 - 2 (2)
2x - 3y + z - 2x + 2y - 4z = 2 - 4
-y - 3z = -2
and
(2x - 3y + z) - 2 (x + 2y - 3z) = 2 - 2 (4)
2x - 3y + z - 2x - 4y + 6z = 2 - 8
-7y + 7z = -6
In augmented matrix form, this step yields
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & -7 & 7 & -6 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%200%20%26%20-1%20%26%20-3%20%26%20-2%20%5C%5C%200%20%26%20-7%20%26%207%20%26%20-6%20%5Cend%7Barray%7D%20%5Cright%5D)
I'll omit these details in the remaining steps.
Eliminate y from the last row by combining -7 (row 2) and row 3 :
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & 0 & 28 & 8 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%200%20%26%20-1%20%26%20-3%20%26%20-2%20%5C%5C%200%20%26%200%20%26%2028%20%26%208%20%5Cend%7Barray%7D%20%5Cright%5D)
Multiply the last row by 1/28 :
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & 0 & 1 & 2/7 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%200%20%26%20-1%20%26%20-3%20%26%20-2%20%5C%5C%200%20%26%200%20%26%201%20%26%202%2F7%20%5Cend%7Barray%7D%20%5Cright%5D)
Eliminate z from the second row by combining 3 (row 3) and row 2 :
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & 0 & -8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%200%20%26%20-1%20%26%200%20%26%20-8%2F7%20%5C%5C%200%20%26%200%20%26%201%20%26%202%2F7%20%5Cend%7Barray%7D%20%5Cright%5D)
Multiply the second row by -1 :
![\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%20-3%20%26%201%20%26%202%20%5C%5C%200%20%26%201%20%26%200%20%26%208%2F7%20%5C%5C%200%20%26%200%20%26%201%20%26%202%2F7%20%5Cend%7Barray%7D%20%5Cright%5D)
Eliminate y and z from the first row by combining 3 (row 2), -1 (row 3), and row 1 :
![\left[ \begin{array}{ccc|c} 2 & 0 & 0 & 36/7 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%202%20%26%200%20%26%200%20%26%2036%2F7%20%5C%5C%200%20%26%201%20%26%200%20%26%208%2F7%20%5C%5C%200%20%26%200%20%26%201%20%26%202%2F7%20%5Cend%7Barray%7D%20%5Cright%5D)
Multiply the first row by 1/2 :
![\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 18/7 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%201%20%26%200%20%26%200%20%26%2018%2F7%20%5C%5C%200%20%26%201%20%26%200%20%26%208%2F7%20%5C%5C%200%20%26%200%20%26%201%20%26%202%2F7%20%5Cend%7Barray%7D%20%5Cright%5D)