# Are there two types of 'slack variables'?

In simplex algorithm, in order to handle inequation constraints, we need to convert them into equations by introducing so-called 'slack variables', like $$\mathbf{a}^T\mathbf{x} + b \leq 0\quad\rightarrow\quad \mathbf{a}^T\mathbf{x} + b + s = 0$$

On the other hand, if some constraints could be violated, we can introduce 'slack variables' to make them 'soft constraints', like $$\mathbf{a}^T\mathbf{x} + b \leq 0\quad\rightarrow\quad \mathbf{a}^T\mathbf{x} + b - s \leq 0$$

These two scenarios are different. In the simplex scenario, it is the equation constraint that is slacked to be an inequation, and in the second scenario, it is the inequation that is slacked. But in both scenarios, we say $$s$$ is the slack variable of constraint $$\mathbf{a}^T\mathbf{x} + b \leq 0$$.

It seems a little bit strange to me , so I wonder if I am wrong somewhere.