Linear programming worksheet

The constraints cut out a polygon, and a linear objective takes its largest value at a corner of it. Find every corner, which means the intercepts and where the two slanted lines cross, evaluate P at each, and keep the biggest.



Sigma Prep · sigmaprep.io/worksheets Name Linear Programming Date Period Find the maximum and where it happens. Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 45$ , $4 x + 2 y \leq 40$ Maximize $P = 3 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 2 y \leq 30$ , $x + 2 y \leq 23$ Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 22$ , $3 x + 3 y \leq 45$ Maximize $P = 2 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 60$ , $x + 4 y \leq 36$ Maximize $P = 3 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 39$ , $3 x + 2 y \leq 37$ Maximize $P = 2 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 56$ , $4 x + 4 y \leq 64$ Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 52$ , $2 x + y \leq 20$ Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 26$ , $3 x + 3 y \leq 42$ Maximize $P = 3 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 36$ , $4 x + y \leq 32$ Maximize $P = 5 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + y \leq 16$ , $x + 2 y \leq 24$ Maximize $P = 3 x + 6 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 31$ , $x + 3 y \leq 25$ Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 34$ , $3 x + 2 y \leq 36$ Maximize $P = x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 25$ , $3 x + y \leq 27$ Maximize $P = 9 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 53$ , $3 x + 3 y \leq 45$ Maximize $P = 4 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 40$ , $3 x + 3 y \leq 48$ Maximize $P = 7 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 20$ , $3 x + 3 y \leq 42$ Maximize $P = 8 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 23$ , $3 x + y \leq 29$ Maximize $P = 4 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 24$ , $4 x + 4 y \leq 48$ Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 28$ , $x + 4 y \leq 35$ Maximize $P = 6 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 45$ , $2 x + 3 y \leq 38$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Linear Programming

Date Period

Find the maximum and where it happens.

Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 45$ , $4 x + 2 y \leq 40$
Maximize $P = 3 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 2 y \leq 30$ , $x + 2 y \leq 23$
Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 22$ , $3 x + 3 y \leq 45$
Maximize $P = 2 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 60$ , $x + 4 y \leq 36$
Maximize $P = 3 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 39$ , $3 x + 2 y \leq 37$
Maximize $P = 2 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 56$ , $4 x + 4 y \leq 64$
Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 52$ , $2 x + y \leq 20$
Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 26$ , $3 x + 3 y \leq 42$
Maximize $P = 3 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 36$ , $4 x + y \leq 32$
Maximize $P = 5 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + y \leq 16$ , $x + 2 y \leq 24$
Maximize $P = 3 x + 6 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 31$ , $x + 3 y \leq 25$
Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 34$ , $3 x + 2 y \leq 36$
Maximize $P = x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 25$ , $3 x + y \leq 27$
Maximize $P = 9 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 53$ , $3 x + 3 y \leq 45$
Maximize $P = 4 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 40$ , $3 x + 3 y \leq 48$
Maximize $P = 7 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 20$ , $3 x + 3 y \leq 42$
Maximize $P = 8 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 23$ , $3 x + y \leq 29$
Maximize $P = 4 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 24$ , $4 x + 4 y \leq 48$
Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 28$ , $x + 4 y \leq 35$
Maximize $P = 6 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 45$ , $2 x + 3 y \leq 38$



Sigma Prep · sigmaprep.io/worksheets Name Linear ProgrammingAnswer keyVersion 1 Date Period Find the maximum and where it happens. Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 45$ , $4 x + 2 y \leq 40$ $P = 84$ at $(7, 6)$ Maximize $P = 3 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 2 y \leq 30$ , $x + 2 y \leq 23$ $P = 45$ at $(15, 0)$ Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 22$ , $3 x + 3 y \leq 45$ $P = 77$ at $(11, 0)$ Maximize $P = 2 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 60$ , $x + 4 y \leq 36$ $P = 37$ at $(8, 7)$ Maximize $P = 3 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 39$ , $3 x + 2 y \leq 37$ $P = 77$ at $(7, 8)$ Maximize $P = 2 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 56$ , $4 x + 4 y \leq 64$ $P = 32$ at $(16, 0)$ Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 52$ , $2 x + y \leq 20$ $P = 117$ at $(0, 13)$ Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 26$ , $3 x + 3 y \leq 42$ $P = 112$ at $(0, 14)$ Maximize $P = 3 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 36$ , $4 x + y \leq 32$ $P = 96$ at $(0, 12)$ Maximize $P = 5 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + y \leq 16$ , $x + 2 y \leq 24$ $P = 80$ at $(16, 0)$ Maximize $P = 3 x + 6 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 31$ , $x + 3 y \leq 25$ $P = 75$ at $(25, 0)$ Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 34$ , $3 x + 2 y \leq 36$ $P = 84$ at $(12, 0)$ Maximize $P = x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 25$ , $3 x + y \leq 27$ $P = 13$ at $(7, 6)$ Maximize $P = 9 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 53$ , $3 x + 3 y \leq 45$ $P = 135$ at $(15, 0)$ Maximize $P = 4 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 40$ , $3 x + 3 y \leq 48$ $P = 64$ at $(16, 0)$ Maximize $P = 7 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 20$ , $3 x + 3 y \leq 42$ $P = 112$ at $(0, 14)$ Maximize $P = 8 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 23$ , $3 x + y \leq 29$ $P = 120$ at $(7, 8)$ Maximize $P = 4 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 24$ , $4 x + 4 y \leq 48$ $P = 108$ at $(0, 12)$ Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 28$ , $x + 4 y \leq 35$ $P = 252$ at $(28, 0)$ Maximize $P = 6 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 45$ , $2 x + 3 y \leq 38$ $P = 106$ at $(7, 8)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Linear ProgrammingAnswer keyVersion 1

Date Period

Find the maximum and where it happens.

Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 45$ , $4 x + 2 y \leq 40$ $P = 84$ at $(7, 6)$
Maximize $P = 3 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 2 y \leq 30$ , $x + 2 y \leq 23$ $P = 45$ at $(15, 0)$
Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 22$ , $3 x + 3 y \leq 45$ $P = 77$ at $(11, 0)$
Maximize $P = 2 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 60$ , $x + 4 y \leq 36$ $P = 37$ at $(8, 7)$
Maximize $P = 3 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 39$ , $3 x + 2 y \leq 37$ $P = 77$ at $(7, 8)$
Maximize $P = 2 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 56$ , $4 x + 4 y \leq 64$ $P = 32$ at $(16, 0)$
Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 52$ , $2 x + y \leq 20$ $P = 117$ at $(0, 13)$
Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 26$ , $3 x + 3 y \leq 42$ $P = 112$ at $(0, 14)$
Maximize $P = 3 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 36$ , $4 x + y \leq 32$ $P = 96$ at $(0, 12)$
Maximize $P = 5 x + 3 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + y \leq 16$ , $x + 2 y \leq 24$ $P = 80$ at $(16, 0)$
Maximize $P = 3 x + 6 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 31$ , $x + 3 y \leq 25$ $P = 75$ at $(25, 0)$
Maximize $P = 7 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + 3 y \leq 34$ , $3 x + 2 y \leq 36$ $P = 84$ at $(12, 0)$
Maximize $P = x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 25$ , $3 x + y \leq 27$ $P = 13$ at $(7, 6)$
Maximize $P = 9 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 53$ , $3 x + 3 y \leq 45$ $P = 135$ at $(15, 0)$
Maximize $P = 4 x + y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 4 y \leq 40$ , $3 x + 3 y \leq 48$ $P = 64$ at $(16, 0)$
Maximize $P = 7 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $2 x + y \leq 20$ , $3 x + 3 y \leq 42$ $P = 112$ at $(0, 14)$
Maximize $P = 8 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 23$ , $3 x + y \leq 29$ $P = 120$ at $(7, 8)$
Maximize $P = 4 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 24$ , $4 x + 4 y \leq 48$ $P = 108$ at $(0, 12)$
Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 28$ , $x + 4 y \leq 35$ $P = 252$ at $(28, 0)$
Maximize $P = 6 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 45$ , $2 x + 3 y \leq 38$ $P = 106$ at $(7, 8)$

How to do these

Maximize $P = 3 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + y \leq 6$ , $x + 3 y \leq 12$

Corners: (0, 0), (6, 0), (0, 4), and where the lines cross, (3, 3).
P there: 0, 18, 8, 15.

The answer is $P = 18$ at $(6, 0)$ .

Where students go wrong

Testing the crossing point only. The best corner is often on an axis, and every corner has to be checked.

Also called objective function, feasible region, optimization with constraints or vertices of a feasible region.

Questions about these worksheets

Yes. Every linear programming sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the linear programming questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Testing the crossing point only. The best corner is often on an axis, and every corner has to be checked.

Linear programming worksheet

Linear Programming

Linear ProgrammingAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

Are these linear programming worksheets free?+

Do the linear programming worksheets come with an answer key?+

Are these linear programming worksheets printable?+

What grade are these linear programming worksheets for?+

Can I get a different version of the same linear programming worksheet?+

Is this a free alternative to Kuta Software for linear programming?+

Can I choose how hard the linear programming questions are?+

What do students get wrong with linear programming?+