The element to pivot on is called 'the pivot element'.
The pivot element's row is called 'the pivot row'.
The pivot element's column is called 'the pivot column'.

To pivot on an element, 

Step 1. if the pivot element is not already 1, we divide the pivot row through
by itself to make the pivot element become 1. 

then 

Step 2. We use that 1 to cause every other element in the pivot column to
become 0.

We can skip Step 1 because the pivot element is already 1.  

So we go on to Step 2:

Out to the side or on scratch paper, we write down the pivot row 

        1 3 | 4 

Multiply it through by -3 because -3 is what is needed to add to the 3 under the
pivot element to make it become 0.  When we multiply the pivot row by -3, we
get:

      -3 -9 | -12

Then we write the second row under it and add:

      -3 -9 | -12
       3  6 |   9
     ------------
       0 -3 |  -3

Then we write that in place of the second row:



That completes the pivoting on the first element in the first row, first column.

The idea is to make the pivot element become 1 and then use it to make all the
other elements in the pivot column become 0's.

Edwin