Graphical Solution of a 2-Variable Linear Program

See Chapter 2: Introduction to Linear Programming.

The Acme Bicycle Company Example (see Chapter 1) is summarized below:


  • x₁: mountain bikes to produce (bikes/day)
  • x₂: racers to produce  (bikes/day)

Variable non-negativity:

  • x₁ ≥ 0,  x₂ ≥ 0

Objective Function:

  • Maximize daily profit ($/day):  Max Z =  15 x₁ + 10 x₂


  • Mountain bike production limit (bikes/day): x₁ ≤ 2
  • Racer production limit (bikes/day): x₂ ≤ 3
  • Metal finishing machine production limit (bikes/day): x₁ + x₂ ≤ 4

Note that units are specified for everything: variables, the objective function, and the constraints. This is important.

This problem has only two variables so it can be sketched on a plane. The animation below graphs the problem in order to find the point that maximizes the daily profit rate, i.e. the optimum point.

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