Converting Inequalities to Equalities with Slack Variables

The standard form of a linear program requires every constraint to be an equality. To convert a $\leq$ inequality into an equality, we introduce a nonnegative slack variable that absorbs the gap between the two sides.

For a constraint of the form

$$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq b,$$

we introduce a slack variable $s \geq 0$ and rewrite the constraint as

$$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n + s = b.$$

The variable $s$ measures how much slack remains between the left-hand side and the bound $b$. Since the original inequality says LHS is at most $b$, the gap $s = b - \text{LHS}$ is automatically nonnegative.

Illustration. The constraint $2x_1 + x_2 \leq 8$ becomes

$$2x_1 + x_2 + s_1 = 8, \qquad s_1 \geq 0.$$

At the point $(x_1, x_2) = (3, 1)$, the LHS equals $7$, so the slack is $s_1 = 8 - 7 = 1$.