slack variable

In mathematics, a slack variable is a non-negative variable that is added to an inequality constraint to convert it into an equality. This is a key step in the simplex method, an algorithm used to solve optimization problems, particularly in linear programming.1

For example, if you have the constraint (x_1 \le 8), you can introduce a slack variable, say (x_3), to make it an equality: (x_1 + x_3 = 8). Here, (x_3) represents the "slack" or unused amount of the resource or capacity defined by the original inequality. The slack variable must be greater than or equal to zero ((x_3 \ge 0)). This transformation allows for algebraic manipulation and systematic exploration of potential solutions, typically at the "extreme points" of the feasible region.12