Solve the following system of linear equations using matrices by row operations: $$\begin{array}{rlrl}2x-3y+z& =9-x+4y-z& =-73x-2y+2z& =12\end{array}$$

Step 1 ：Step 1: Write the system of equations in matrix form: $$\left[\begin{array}{ccccccccccccc}2& -3& 1& |& 9-1& 4& -1& |& -73& -2& 2& |& 12\end{array}\right]$$

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: $$\left[\begin{array}{ccccccccccccc}2& -3& 1& |& 90& 1& 0& |& 20& 0.5& 0.5& |& -1.5\end{array}\right]$$

Step 3 ：Step 3: Multiply the third row by 2 to get rid of the 0.5: $$\left[\begin{array}{ccccccccccccc}2& -3& 1& |& 90& 1& 0& |& 20& 0& 1& |& -3\end{array}\right]$$

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: $$\left[\begin{array}{ccccccccccccc}2& 0& 0& |& 00& 1& 0& |& 20& 0& 1& |& -3\end{array}\right]$$

Step 5 ：Step 5: Divide the first row by 2: $$\left[\begin{array}{ccccccccccccc}1& 0& 0& |& 00& 1& 0& |& 20& 0& 1& |& -3\end{array}\right]$$