The formula
max cᵀx subject to Ax ≤ b, x ≥ 0
Why it works
The simplex method rests on a geometric fact that changes everything: linear constraints enclose a region shaped like a polyhedron, and the optimum of a linear function over that region always sits at a vertex. That turns a problem with infinitely many candidate points into one with finitely many. The algorithm starts at a vertex and keeps hopping to the neighbour that most improves the objective, stopping when none does: that vertex is then the optimum. Bland’s rule, which always picks the first eligible column, guarantees the method never cycles.
How to solve it by hand
- Write the objective and the constraints with one coefficient per variable
- Add a slack variable to each constraint to turn the inequality into an equality
- The variable that most improves the objective enters and the first to run out leaves
- Repeat until no column improves the result
What is worth knowing
The standard form this page solves needs less-than-or-equal constraints with positive limits and non-negative variables, which covers ninety per cent of textbook problems. A greater-than-or-equal constraint, or a negative limit, would need the two-phase simplex, and here you get a warning rather than a wrong answer. The slack column is the most useful part of the result and the least looked at: it says how much of each resource is left over at the optimum, and the constraints that run out entirely are the only ones actually limiting the result. Loosening any other one buys you nothing.