Why do we have to add slack variables and artificial variables in Simplex method at Linear programming?

Question

I learn why we have to use these variable in lectures.

• Slack variable : Make linear inequalities to linear equalities

• Artificial variable : Know whether the basic feasible solution exist or not

But I think that without these variable, we can find the vertex of the convex polytope by using Gaussian elimination. We just convert the inequalities into equalities without slack variables.

Also, by using these equalities and Gaussian elimination, we know whether there are solutions or not.

Why do we have to add these variable? Where is the lack of my understanding?

Answer 1

Let's add the constraint, $x \ge 0$, it doesn't change the feasible set but if you just ignore the inequality and set it to equality, you would not get a feasible solution.

Hence, we have to decide which set of constraints to set to be active.