Practice Linear Programming in Math

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

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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+2yC = 3x + 2y subject to x+y≥5x+y\ge 5, 2x+y≥82x+y\ge 8, x,y≥0x,y\ge 0.

Example 3

medium
A shop profits $4\$4 per xx and $5\$5 per yy. 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+3yP = 2x + 3y over corners (0,0)(0,0), (4,0)(4,0), (0,3)(0,3), (2,2)(2,2).

Example 6

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

Example 7

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

Example 8

easy
Write a constraint: a worker has at most 88 hours, using 22 hours per unit of xx.

Example 9

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

Example 10

easy
A constraint says production must be at least 1010 units of xx. Write it.

Example 11

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 12

medium
Find the feasible vertex of x+y≤6x+y\le 6 and 2x+y≤82x+y\le 8 in the first quadrant where the two lines meet.

Example 13

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

Example 14

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

Example 15

challenge
Maximize P=3x+4yP = 3x + 4y over x+y≤4x+y\le 4, x+3y≤6x+3y\le 6, x,y≥0x,y\ge0. Find the optimum.

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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