Why are we allowed to add slack or surplus variables in linear programming?

Question

It may sound like a silly question, but I just realized that I can do the simplex algorithm, yet I can't answer that question. When we introduce slack or surplus variables, why don't we 'undo' that step at the end? Why does their presence not change the answer?

Answer 1

A slack variable does not change the problem, just its representation. If you have a constraint $$ f(x) \le b$$ with $x$ variables and $b$ a constant, then you replace it with $$ f(x)+ \lambda=b$$ and $$ \lambda \ge 0$$ which amounts to the same thing. $\lambda$ is a new variable called a slack variable (cause it "picks up the slack" of the original inequality).

Thus introducing slack variables just gives you a different representation of the same problem. There's no reason to "get rid of" them... they're essential in the new representation.

At the end of the day you have your optimum $(x,\lambda).$ $x$ is your optimum answer and then the value for $\lambda$ just tells you how much room that constraint still has at the optimum $x$... if it is saturated to $\lambda = 0$ that means the constraint binds.