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.

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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: Linear programming maximizes or minimizes a linear objective over a region cut out by linear constraints, and the optimum always sits at a corner of that region.

Common stuck point: 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.

Sense of Study hint: Ask: Am I optimizing a linear objective over a region bounded by linear inequalities, with the answer expected at a corner?

Worked Examples

Example 1

medium

Maximize

z = 3x + 2y

subject to

x + y \leq 4

x \geq 0

y \geq 0

Answer

z = 12

(4, 0)

First step

Step 1: Identify the feasible region: the triangle with vertices

(0,0)

(4,0)

(0,4)

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Example 2

hard

Minimize

z = 2x + 5y

subject to

x + 2y \geq 6

x + y \geq 4

x \geq 0

y \geq 0

Example 3

medium

Maximize

P = 5x + 6y

subject to

x + y \le 8

2x + y \le 12

x, y \ge 0

Example 4

medium

Find the feasible vertex of

x+y\le 6

and

2x+y\le 8

in the first quadrant where the two lines meet.

Example 5

hard

A farmer plants wheat (

x

acres,

\$200

profit/acre) and corn (

y

acres,

\$300

profit/acre). Wheat needs

1

hour labor/acre; corn needs

2

hours. Total land

\le 50

acres; labor

\le 80

hours. Find the best mix.

Example 6

hard

Show that if the objective is parallel to a binding constraint, there can be infinitely many optimal solutions.

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.

Example 3

easy

Evaluate the objective

P = 3x + 2y

at the corner point

(2, 1)

Example 4

easy

Does the point

(1, 1)

satisfy the constraint

x + y \le 5

Example 5

easy

(6, 0)

feasible under

x + y \le 5

with

x,y \ge 0

Example 6

easy

At which of

(0,0)

and

(3,3)

P = x + y

larger?

Example 7

easy

State the non-negativity constraints for a problem producing

x

chairs and

y

tables.

Example 8

easy

Write a constraint: a worker has at most

8

hours, using

2

hours per unit of

x

Example 9

easy

A constraint says production must be at least

10

units of

x

. Write it.

Example 10

easy

How many corner points does a triangular feasible region have to check?

Example 11

medium

Maximize

P = 2x + 3y

over corners

(0,0)

(4,0)

(0,3)

(2,2)

Example 12

medium

Minimize

C = x + 2y

over corners

(1,4)

(3,1)

(5,2)

Example 13

medium

Find the intersection of

x + y = 4

and

x - y = 2

(a corner point).

Example 14

medium

A shop profits

\$4

per

x

and

\$5

per

y

. Write the objective to maximize.

Example 15

medium

Region:

x \ge 0

y \ge 0

x + y \le 6

. Maximize

P = x + y

Example 16

medium

Is the feasible region for

x \ge 0

y \ge 0

x + y \ge 2

bounded or unbounded?

Example 17

medium

Maximize

P = 5x + 4y

subject to

x \le 4

y \le 3

x,y \ge 0

Example 18

medium

A constraint

2x + 3y \le 12

meets the

x

-axis where?

Example 19

challenge

Maximize

P = 3x + 4y

over

x+y\le 4

x+3y\le 6

x,y\ge0

. Find the optimum.

Example 20

challenge

A factory makes

x,y

with profit

P=40x+30y

, limits

x+y\le 8

(labor),

2x+y\le 10

(material),

x,y\ge0

. Maximize.

Example 21

challenge

Why must the optimum of a linear objective on a bounded polygon occur at a vertex?

Example 22

medium

Maximize

P = x + 2y

over corners

(0,0)

(0,5)

(4,3)

(6,0)

Example 23

easy

Evaluate

P = 4x + 3y

at the corner

(2, 5)

Example 24

easy

(3, 4)

feasible for

x + 2y \le 12

x,y\ge 0

Example 25

easy

Find the corner of the region

x \ge 0

y\ge 0

x + y \le 10

that maximizes

P = 2x + 5y

Example 26

easy

Find where

3x + 4y = 24

crosses the

y

-axis.

Example 27

easy

Minimize

C = x + y

over corners

(0,2)

(3,0)

(2,1)

Example 28

medium

Find the intersection of

2x + y = 10

and

x + y = 6

Example 29

medium

A bakery makes muffins (

\$2

profit each) and scones (

\$3

profit each). Each muffin uses

1

oz of butter; each scone uses

2

oz. They have

20

oz of butter and want at most

15

items total. Write the LP.

Example 30

medium

Maximize

P = 2x + 3y

subject to

x + 2y\le 20

x + y \le 15

x,y\ge 0

Example 31

medium

Minimize

C = 3x + 2y

subject to

x+y\ge 5

2x+y\ge 8

x,y\ge 0

Example 32

medium

Maximize

P = 6x + 4y

subject to

x + y \le 10

2x + 3y \le 24

x,y\ge 0

Example 33

medium

Maximize

P = x + y

over the region

x+y \le 4

x,y \ge 0

Example 34

medium

Find where

4x + 3y = 24

meets the

x

-axis.

Example 35

medium

(5, 4)

feasible for

3x + 2y\le 25

x + 4y \le 20

x, y\ge 0

Example 36

medium

At which corner is

P = 4x + y

maximized over

(0,0)

(5,0)

(3,4)

(0,6)

Example 37

hard

Maximize

P = 7x + 9y

subject to

x+y\le 6

3x+y\le 12

x+2y\le 10

x,y\ge 0

Example 38

hard

A diet problem: each unit of food A costs

\$3

and provides

4

g of protein and

1

g of fat; each unit of food B costs

\$2

and provides

2

g of protein and

1

g of fat. Need at least

20

g protein and at most

8

g fat. Minimize cost.

Example 39

hard

Maximize

P = 3x + 5y

on the region

x+y\le 4

2x+y\le 6

x,y\ge 0

Example 40

challenge

A factory produces three products with profits

\$10, \$12, \$15

per unit and machine-hour costs

1, 2, 3

hours per unit. With

24

machine-hours available and a demand cap of

10

units per product, but no constraint linking products, what is the best single-product plan?

Example 41

challenge

Minimize

C = 4x + 5y

subject to

x + y \ge 6

x + 3y \ge 9

x, y\ge 0

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Related Concepts

Inequalities Systems of Equations Constraint System

Background Knowledge

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

inequalitiessystems of equationsconstraint system