Linear Programming Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Linear Programming.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The optimal solution to a linear program always occurs at a vertex (corner point) of the feasible region — never in the interior.

Common stuck point: Students optimize outside the feasible region or forget to include all constraints when finding corner points.

Sense of Study hint: Graph constraints first, shade the feasible region, then test corner points.

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Identify the feasible region: the triangle with vertices (0,0), (4,0), (0,4).
  2. 2
    Step 2: Evaluate z at each vertex: z(0,0) = 0, z(4,0) = 12, z(0,4) = 8.
  3. 3
    Step 3: Maximum is z = 12 at (4, 0).
  4. 4
    Check: The coefficient of x is larger, so the maximum favors x ✓

Answer

z = 12 at (4, 0)
Linear programming optimizes a linear objective over a feasible region defined by linear inequalities. The optimal value always occurs at a vertex (corner point) of the feasible region.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A feasible region has vertices at (0,0), (5,0), (3,4), (0,6). Maximize z = x + 2y.

Example 2

medium
A company makes chairs (\40 profit) and tables (\70 profit). Each chair takes 2 hours, each table takes 5 hours. With 40 hours available, maximize profit.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

inequalitiessystems of equationsconstraint system