Linear Programming and Simplex

Key idea

Optimise linear objectives.

In depth

An LP optimises a linear objective under linear constraints; the optimum is at a vertex of the feasible polytope. The simplex method moves from vertex to vertex improving the objective until none improves.

Worked example

Maximise 3x + 2y subject to x + y ≤ 4, x ≤ 3: optimum at (3, 1) with value 11.

Real-world uses

• Planning production, transport and staff rotas to maximise profit or minimise cost

Common mistake

Forgetting to convert inequalities to equalities using slack variables.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Differential Equations, Abstract Algebra & Applied Math

• Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
• Runge–Kutta Methods – Higher-accuracy numerical solvers.
• Second-Order Homogeneous Equations – Use the characteristic equation.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.