Practice Linear Programming in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

medium
Minimize C=3x+2y subject to x+y≥5, 2x+y≥8, x,y≥0.

Example 3

medium
A shop profits $4 per x and $5 per y. Write the objective to maximize.

Example 4

medium
True or False: A linear objective on a bounded feasible polygon always achieves its maximum at a vertex.

Example 5

medium
Maximize P=2x+3y over corners (0,0), (4,0), (0,3), (2,2).

Example 6

medium
Is the feasible region for x≥0, y≥0, x+y≥2 bounded or unbounded?

Example 7

challenge
Minimize C=4x+5y subject to x+y≥6, x+3y≥9, x,y≥0.

Example 8

easy
Write a constraint: a worker has at most 8 hours, using 2 hours per unit of x.

Example 9

medium
Maximize P=2x+3y subject to x+2y≤20, x+y≤15, x,y≥0.

Example 10

easy
A constraint says production must be at least 10 units of x. Write it.

Example 11

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

Example 12

medium
Find the feasible vertex of x+y≤6 and 2x+y≤8 in the first quadrant where the two lines meet.

Example 13

medium
Maximize P=x+2y over corners (0,0), (0,5), (4,3), (6,0).

Example 14

medium
Maximize P=x+y over the region x+y≤4, x,y≥0.

Example 15

challenge
Maximize P=3x+4y over x+y≤4, x+3y≤6, x,y≥0. Find the optimum.

Example 16

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

Example 17

easy
Evaluate P=4x+3y at the corner (2,5).

Example 18

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

Example 19

easy
Does the point (1,1) satisfy the constraint x+y≤5?

Example 20

challenge
Why must the optimum of a linear objective on a bounded polygon occur at a vertex?
← Back to Linear Programming Worked Examples Formula Explained