[ 1  1 1 |-1]
[-1  2 1 | 6] 
[ 2 -1 3 |15]

R1 + R2→R2  means to add row 1 and row 2 and take what you get and
replace row 2 with it.

So we add row 1 and row 2

                     R1 = [  1  1 1 |-1]
                   + R2 = [ -1  2 1 | 6]
                    --------------------   
                  R1+R2 = [  0  3 2 | 5]

Now we replace row 2 with that sum:

[1  1 1 |-1]
[0  3 2 | 5]
[2 -1 3 |15]

-2R1+R3→R3  means to multiply R1 by -2 and add it to row 3
and take what you get and replace row 3 by it:

So we multiply row 1 by -2 

                     R1 = [  1  1  1 |-1]
                                    ×(-2)
                     -------------------
                   -2R1 = [ -2 -2 -2 | 2]
add to row 3-->      R3 = [  2 -1  3 |15] 
                     --------------------
                 -2R+R3 = [  0 -3  1 |17]

Now we replace row 3 by that result:

[1  1 1 |-1]
[0  3 2 | 5]
[0 -3 1 |17]

Edwin