First I name the matrix of our system.

mA={{3,0,-1,1},{1,1,-2,2},{2,-1,1,3}}

To display this matrix in the traditional form I use

MatrixForm[mA]

Now I row reduce the matrix mA. The command that performs row reduction is applied to the matrix mA and then, MatrixForm displays the result in convenient form:

MatrixForm[RowReduce[mA]]

What I see is that each row has a pivot entry. This tells me that whatever the augmented column is, the system with this matrix will always have a solution. The reason for this is that a system with three equation can have at most three pivots. Here, all pivots are in the columns which belong to the system, so the augmented column cannot be a pivot column.

Now form the augmented matrix

mAa={{3,0,-1,1,-2},{1,1,-2,2,3},{2,-1,1,3,8}}

MatrixForm[mAa]

MatrixForm[RowReduce[mAa]]