In terms of programming. Like, lets say you have F1+F2+F3+15=0 and you move the 15 to the other side, just as you're supposed to, which would be F1+F2+F3= -15. Is that what they mean by pivoting? The code moves the 15?
No. The "pivot" is a single entry in a row of a matrix. "Pivoting" refers to a process involving several steps. The row is divided by the value of the pivot to turn the pivot into 1. Then all the entries in the column of the pivot and directly below it are turned into zeroes by subtracting some multiple of the pivot's row from them.
If you're planning to write a serious program to solve linear equations, you should consult a text on numerical methods. The better numerical algorithms do not simply march down the main diagonal using each entry as a pivot.
How would I call it then - to get each equation into the right format?
I need it for reference for Gaussian elimination systems.
What do you mean by "it"? Do you mean the example, you gave? What you did (as best I can determine) is not a process that has a distinctive name. You transformed the equation into one that has no constant on the left hand side.
I can't tell whether you are merely interested in terminology for something or whether you want to know how to write a computer program to do Gaussian elimination.