Linear Programming Formula

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

The Formula

max⁡/min⁡  cTx  subject to  Ax≤b\max/\min\; c^Tx\; \text{subject to}\; Ax \le b

When to use: You search the corners of an allowed region for the best score.

Quick Example

max⁡z=3x+2y\max z = 3x+2y subject to x+y≤4,  x≥0,  y≥0x+y \le 4,\; x \ge 0,\; y \ge 0 — optimal at a corner.

Notation

max⁡z\max z or min⁡z\min z with linear constraints.

What This Formula Means

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

You search the corners of an allowed region for the best score.

Formal View

Find x∈Rnx\in\mathbb{R}^n that optimizes cTxc^Tx over a polyhedral feasible set {x∣Ax≤b}\{x \mid Ax \le b\}.

Worked Examples

Example 1

medium
Maximize z=3x+2yz = 3x + 2y subject to x+y≤4x + y \leq 4, x≥0x \geq 0, y≥0y \geq 0.

Answer

z=12z = 12 at (4,0)(4, 0)

First step

1
Step 1: Identify the feasible region: the triangle with vertices (0,0)(0,0), (4,0)(4,0), (0,4)(0,4).

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Example 2

hard
Minimize z=2x+5yz = 2x + 5y subject to x+2y≥6x + 2y \geq 6, x+y≥4x + y \geq 4, x≥0x \geq 0, y≥0y \geq 0.

Example 3

medium
Maximize P=5x+6yP = 5x + 6y subject to x+y≤8x + y \le 8, 2x+y≤122x + y \le 12, x,y≥0x, y \ge 0.

Common Mistakes

  • Searching the interior for the best value - the optimum of a linear objective lies at a vertex of the feasible region; evaluate the objective at each corner
  • Forgetting hidden constraints like x≥0x\ge 0 - real quantities are often nonnegative, which changes the corners
  • Mixing up maximize and minimize - read whether you want the largest or smallest objective value before comparing corners

Why This Formula Matters

Linear programming is the bridge from systems of inequalities to real optimization — it tells students that with linear limits the best outcome is never in the middle but at a vertex, which is the whole reason graphing the feasible region and testing corners works. Recognizing it by "Am I optimizing a linear objective over a region bounded by linear inequalities, with the answer expected at a corner?" — rather than by familiar numbers — is what lets a student tell it apart from systems of equations and graphing inequalities and single-variable optimization in a mixed problem set.

Frequently Asked Questions

What is the Linear Programming formula?

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

How do you use the Linear Programming formula?

You search the corners of an allowed region for the best score.

What do the symbols mean in the Linear Programming formula?

max⁡z\max z or min⁡z\min z with linear constraints.

Why is the Linear Programming formula important in Math?

Linear programming is the bridge from systems of inequalities to real optimization — it tells students that with linear limits the best outcome is never in the middle but at a vertex, which is the whole reason graphing the feasible region and testing corners works. Recognizing it by "Am I optimizing a linear objective over a region bounded by linear inequalities, with the answer expected at a corner?" — rather than by familiar numbers — is what lets a student tell it apart from systems of equations and graphing inequalities and single-variable optimization in a mixed problem set.

What do students get wrong about Linear Programming?

The procedure for linear programming is the easy part; the trap is searching the interior for the best value. Asking "Am I optimizing a linear objective over a region bounded by linear inequalities, with the answer expected at a corner?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Linear Programming formula?

Before studying the Linear Programming formula, you should understand: inequalities, systems of equations, constraint system.

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