Solve the following system of linear equations using matrices by row operations: \[ \begin{align*} 2x - 3y + z &= 9\ -x + 4y - z &= -7\ 3x - 2y + 2z &= 12 \end{align*} \]

Step 1 ：Step 1: Write the system of equations in matrix form: \[ \begin{bmatrix} 2 & -3 & 1 & | & 9\ -1 & 4 & -1 & | & -7\ 3 & -2 & 2 & | & 12 \end{bmatrix} \]

Step 2 ：Step 2: Use row operations to convert the matrix to row echelon form. Add the first row to the second row to eliminate the -1 in the second row. Subtract 1.5 times the first row from the third row to eliminate the 3 in the third row: \[ \begin{bmatrix} 2 & -3 & 1 & | & 9\ 0 & 1 & 0 & | & 2\ 0 & 0.5 & 0.5 & | & -1.5 \end{bmatrix} \]

Step 3 ：Step 3: Multiply the third row by 2 to get rid of the 0.5: \[ \begin{bmatrix} 2 & -3 & 1 & | & 9\ 0 & 1 & 0 & | & 2\ 0 & 0 & 1 & | & -3 \end{bmatrix} \]

Step 4 ：Step 4: Add 3 times the third row to the first row to get rid of the 1 in the first row. Subtract the second row from the first row to get rid of the -3 in the first row: \[ \begin{bmatrix} 2 & 0 & 0 & | & 0\ 0 & 1 & 0 & | & 2\ 0 & 0 & 1 & | & -3 \end{bmatrix} \]

Step 5 ：Step 5: Divide the first row by 2: \[ \begin{bmatrix} 1 & 0 & 0 & | & 0\ 0 & 1 & 0 & | & 2\ 0 & 0 & 1 & | & -3 \end{bmatrix} \]