Slack Variables and Standard Form - Simplex, Pivot by Pivot

Simplex starts by rewriting inequalities as equalities with slack variables. This lesson shows how unused capacity becomes a variable in each constraint row. The exact matrix is the standard-form system the tableau will extend.

highlighted = computed this step

Slack variables

A slack variable records unused capacity. The first row becomes 2x plus 1y plus 1s1=4. Why: an inequality becomes an equality once the unused part is named.

2x+{}1y+{}1s_{1}=4

Second constraint

The second row becomes 1x plus 2y plus 1s2=4. Why: each constraint gets its own slack, so each row keeps its own accounting.

1x+{}2y+{}1s_{2}=4

Feasible variables

Feasibility now means every decision and slack variable is nonnegative. Why: a negative slack would claim more resource was used than the constraint allows.

\text{all variables are nonnegative}

Diagram note

The matrix shows the exact equality system that simplex will use as its starting point. Pixel positions are rounded for layout; every number shown is exact.

\text{slacks turn inequalities into equalities}