What is a slack variable used for in linear programming?

A slack variable is specifically introduced in linear programming to transform "less than or equal to" (≤) constraints into equalities. This is crucial because the methods used to solve linear programming problems, such as the simplex method, require that all constraints be expressed as equalities in order to create a feasible solution space.

When a constraint is a ≤ type, a slack variable is added to the left side of the equation, allowing the constraint to be satisfied while maintaining the mathematical balance. For instance, if you have a constraint like ( x_1 + x_2 \leq 10 ), you can add a slack variable ( s ) to turn it into ( x_1 + x_2 + s = 10 ), where ( s \geq 0 ). This means that some amount of "slack" can be unused in the solution, making it easier to handle during optimization.

In contrast, other options do not accurately describe the function of slack variables in linear programming. They do not play a role in eliminating variables, introducing randomness, or directly minimizing the objective function. Their main purpose is to maintain the integrity of inequality constraints while making the problem more tractable for solution techniques.