Section 3-4

Linear Programming

Solving for Minimums or Maximums with constraints

• Graph the inequalities(CONSTRAINTS) & shade• Identify the overlapping region• Identify the corners of the region• Test the corners to identify the max or min

Maximums & Minimums

• If a maximum or minimum exists, they will occur at a corner point.

Example

• Given:

• Find the point (x,y) thatMinimizes C=5x+3y

Example

• Given:

• Find the point (x,y) thatMaximizes P=3x+2y

ExampleMachine Console

(X)Portable(Y)

Hours Available

A 1 h 2 h 16

B 1 h 1 h 9C 4 h 1 h 24

Constraints:

ExampleMachine Console

(X)Portable(Y)

Hours Available

A 1 h 2 h 16

B 1 h 1 h 9C 4 h 1 h 24

Constraints:

MAXIMIZE PROFIT:

P=60x+40y

ExampleBrand X Brand Y Max daily

Need

Vitamin A 2 mg 1 mg 4 mg

Vitamin B 3 mg 4 mg 11 mgVitamin C 25 mg 50 mg 100 mgCost Per Pill 6 ¢ 8 ¢

Constraints:

MINIMIZE COST: