A shop makes bracelets (x) and necklaces (y). Each bracelet earns $7 and each necklace $10. Each bracelet takes 4 minutes of beading and 1 minute of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 16 minutes of beading and 6 minutes of packing available. Write the constraints and the profit, then maximize it.

4x + 2y ≤ 16, x + y ≤ 6, x ≥ 0, y ≥ 0; P = 7x + 10y; maximum P = $60 at (0, 6)

A shop makes bookshelves (x) and desks (y). Each bookshelf earns $15 and each desk $11. Each bookshelf takes 4 hours of sanding and 4 hours of painting; each desk takes 2 hours of sanding and 3 hours of painting. There are 12 hours of sanding and 14 hours of painting available. Write the constraints and the profit, then maximize it.

4x + 2y ≤ 12, 4x + 3y ≤ 14, x ≥ 0, y ≥ 0; P = 15x + 11y; maximum P = $52 at (2, 2)

A shop makes tables (x) and chairs (y). Each table earns $15 and each chair $6. Each table takes 4 hours of carpentry and 1 hour of finishing; each chair takes 1 hour of carpentry and 1 hour of finishing. There are 40 hours of carpentry and 16 hours of finishing available. Write the constraints and the profit, then maximize it.

4x + y ≤ 40, x + y ≤ 16, x ≥ 0, y ≥ 0; P = 15x + 6y; maximum P = $168 at (8, 8)

A shop makes bracelets (x) and necklaces (y). Each bracelet earns $11 and each necklace $5. Each bracelet takes 1 minute of beading and 4 minutes of packing; each necklace takes 1 minute of beading and 1 minute of packing. There are 11 minutes of beading and 26 minutes of packing available. Write the constraints and the profit, then maximize it.

x + y ≤ 11, 4x + y ≤ 26, x ≥ 0, y ≥ 0; P = 11x + 5y; maximum P = $85 at (5, 6)

A shop makes small boxes (x) and large boxes (y). Each small box earns $6 and each large box $14. Each small box takes 3 minutes of cutting and 4 minutes of folding; each large box takes 1 minute of cutting and 4 minutes of folding. There are 9 minutes of cutting and 20 minutes of folding available. Write the constraints and the profit, then maximize it.

3x + y ≤ 9, 4x + 4y ≤ 20, x ≥ 0, y ≥ 0; P = 6x + 14y; maximum P = $70 at (0, 5)

A shop makes cakes (x) and pies (y). Each cake earns $11 and each pie $13. Each cake takes 2 hours of oven time and 3 hours of decorating; each pie takes 2 hours of oven time and 1 hour of decorating. There are 24 hours of oven time and 20 hours of decorating available. Write the constraints and the profit, then maximize it.

2x + 2y ≤ 24, 3x + y ≤ 20, x ≥ 0, y ≥ 0; P = 11x + 13y; maximum P = $156 at (0, 12)

A shop makes bracelets (x) and necklaces (y). Each bracelet earns $6 and each necklace $3. Each bracelet takes 1 minute of beading and 3 minutes of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 19 minutes of beading and 22 minutes of packing available. Write the constraints and the profit, then maximize it.

x + 2y ≤ 19, 3x + y ≤ 22, x ≥ 0, y ≥ 0; P = 6x + 3y; maximum P = $51 at (5, 7)

Worksheets · Algebra 2

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 the objective at each, and keep the biggest, or the smallest when minimizing.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Linear Programming Date Period Find the maximum or minimum and where it happens. Maximize over a feasible region Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 16$ , $4 x + 2 y \leq 18$ Maximize $P = 9 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 7$ , $4 x + y \leq 6$ Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 16$ , $2 x + y \leq 7$ Maximize $P = 8 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 14$ , $4 x + 4 y \leq 36$ Maximize $P = 2 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 20$ , $x + y \leq 6$ Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 13$ , $x + 4 y \leq 16$ Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 12$ , $x + 2 y \leq 6$ Minimize over a feasible region Minimize $C = 2 x + y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 2 y \geq 18,$ $x + 2 y \geq 15$ Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 4 y \geq 17,$ $4 x + 3 y \geq 18$ Minimize $C = 2 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + y \geq 11,$ $3 x + 3 y \geq 21$ Minimize $C = 5 x + 3 y$ subject to $x \geq 0,$ $y \geq 0,$ $x + 2 y \geq 18,$ $x + y \geq 12$ Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 4 y \geq 18,$ $2 x + y \geq 6$ Minimize $C = 4 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 2 y \geq 26,$ $x + 4 y \geq 22$ Minimize $C = 4 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + y \geq 10,$ $x + 2 y \geq 8$ Solve linear programming word problems A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $7 and each necklace $10. Each bracelet takes 4 minutes of beading and 1 minute of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 16 minutes of beading and 6 minutes of packing available. Write the constraints and the profit, then maximize it. A shop makes bookshelves ( $x$ ) and desks ( $y$ ). Each bookshelf earns $15 and each desk $11. Each bookshelf takes 4 hours of sanding and 4 hours of painting; each desk takes 2 hours of sanding and 3 hours of painting. There are 12 hours of sanding and 14 hours of painting available. Write the constraints and the profit, then maximize it. A shop makes tables ( $x$ ) and chairs ( $y$ ). Each table earns $15 and each chair $6. Each table takes 4 hours of carpentry and 1 hour of finishing; each chair takes 1 hour of carpentry and 1 hour of finishing. There are 40 hours of carpentry and 16 hours of finishing available. Write the constraints and the profit, then maximize it. A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $11 and each necklace $5. Each bracelet takes 1 minute of beading and 4 minutes of packing; each necklace takes 1 minute of beading and 1 minute of packing. There are 11 minutes of beading and 26 minutes of packing available. Write the constraints and the profit, then maximize it. A shop makes small boxes ( $x$ ) and large boxes ( $y$ ). Each small box earns $6 and each large box $14. Each small box takes 3 minutes of cutting and 4 minutes of folding; each large box takes 1 minute of cutting and 4 minutes of folding. There are 9 minutes of cutting and 20 minutes of folding available. Write the constraints and the profit, then maximize it. A shop makes cakes ( $x$ ) and pies ( $y$ ). Each cake earns $11 and each pie $13. Each cake takes 2 hours of oven time and 3 hours of decorating; each pie takes 2 hours of oven time and 1 hour of decorating. There are 24 hours of oven time and 20 hours of decorating available. Write the constraints and the profit, then maximize it. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Linear Programming

Date Period

Find the maximum or minimum and where it happens.

Maximize over a feasible region
Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 16$ , $4 x + 2 y \leq 18$
Maximize $P = 9 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 7$ , $4 x + y \leq 6$
Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 16$ , $2 x + y \leq 7$
Maximize $P = 8 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 14$ , $4 x + 4 y \leq 36$
Maximize $P = 2 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 20$ , $x + y \leq 6$
Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 13$ , $x + 4 y \leq 16$
Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 12$ , $x + 2 y \leq 6$

Minimize over a feasible region
Minimize $C = 2 x + y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 2 y \geq 18,$ $x + 2 y \geq 15$
Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 4 y \geq 17,$ $4 x + 3 y \geq 18$
Minimize $C = 2 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + y \geq 11,$ $3 x + 3 y \geq 21$
Minimize $C = 5 x + 3 y$ subject to $x \geq 0,$ $y \geq 0,$ $x + 2 y \geq 18,$ $x + y \geq 12$
Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 4 y \geq 18,$ $2 x + y \geq 6$
Minimize $C = 4 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 2 y \geq 26,$ $x + 4 y \geq 22$
Minimize $C = 4 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + y \geq 10,$ $x + 2 y \geq 8$

Solve linear programming word problems
A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $7 and each necklace $10. Each bracelet takes 4 minutes of beading and 1 minute of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 16 minutes of beading and 6 minutes of packing available. Write the constraints and the profit, then maximize it.
A shop makes bookshelves ( $x$ ) and desks ( $y$ ). Each bookshelf earns $15 and each desk $11. Each bookshelf takes 4 hours of sanding and 4 hours of painting; each desk takes 2 hours of sanding and 3 hours of painting. There are 12 hours of sanding and 14 hours of painting available. Write the constraints and the profit, then maximize it.
A shop makes tables ( $x$ ) and chairs ( $y$ ). Each table earns $15 and each chair $6. Each table takes 4 hours of carpentry and 1 hour of finishing; each chair takes 1 hour of carpentry and 1 hour of finishing. There are 40 hours of carpentry and 16 hours of finishing available. Write the constraints and the profit, then maximize it.
A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $11 and each necklace $5. Each bracelet takes 1 minute of beading and 4 minutes of packing; each necklace takes 1 minute of beading and 1 minute of packing. There are 11 minutes of beading and 26 minutes of packing available. Write the constraints and the profit, then maximize it.
A shop makes small boxes ( $x$ ) and large boxes ( $y$ ). Each small box earns $6 and each large box $14. Each small box takes 3 minutes of cutting and 4 minutes of folding; each large box takes 1 minute of cutting and 4 minutes of folding. There are 9 minutes of cutting and 20 minutes of folding available. Write the constraints and the profit, then maximize it.
A shop makes cakes ( $x$ ) and pies ( $y$ ). Each cake earns $11 and each pie $13. Each cake takes 2 hours of oven time and 3 hours of decorating; each pie takes 2 hours of oven time and 1 hour of decorating. There are 24 hours of oven time and 20 hours of decorating available. Write the constraints and the profit, then maximize it.



Sigma Prep · sigmaprep.io/worksheets Name Linear ProgrammingAnswer keyVersion 1 Date Period Find the maximum or minimum and where it happens. Maximize over a feasible region Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 16$ , $4 x + 2 y \leq 18$ $P = 31$ at $(4, 1)$ Maximize $P = 9 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 7$ , $4 x + y \leq 6$ $P = 23$ at $(1, 2)$ Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 16$ , $2 x + y \leq 7$ $P = 36$ at $(0, 4)$ Maximize $P = 8 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 14$ , $4 x + 4 y \leq 36$ $P = 72$ at $(9, 0)$ Maximize $P = 2 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 20$ , $x + y \leq 6$ $P = 35$ at $(0, 5)$ Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 13$ , $x + 4 y \leq 16$ $P = 117$ at $(13, 0)$ Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 12$ , $x + 2 y \leq 6$ $P = 24$ at $(0, 3)$ Minimize over a feasible region Minimize $C = 2 x + y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 2 y \geq 18,$ $x + 2 y \geq 15$ $C = 9$ at $(0, 9)$ Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 4 y \geq 17,$ $4 x + 3 y \geq 18$ $C = 30$ at $(3, 2)$ Minimize $C = 2 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + y \geq 11,$ $3 x + 3 y \geq 21$ $C = 14$ at $(7, 0)$ Minimize $C = 5 x + 3 y$ subject to $x \geq 0,$ $y \geq 0,$ $x + 2 y \geq 18,$ $x + y \geq 12$ $C = 36$ at $(0, 12)$ Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 4 y \geq 18,$ $2 x + y \geq 6$ $C = 30$ at $(1, 4)$ Minimize $C = 4 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 2 y \geq 26,$ $x + 4 y \geq 22$ $C = 44$ at $(6, 4)$ Minimize $C = 4 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + y \geq 10,$ $x + 2 y \geq 8$ $C = 28$ at $(4, 2)$ Solve linear programming word problems A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $7 and each necklace $10. Each bracelet takes 4 minutes of beading and 1 minute of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 16 minutes of beading and 6 minutes of packing available. Write the constraints and the profit, then maximize it. $4 x + 2 y \leq 16,$ $x + y \leq 6,$ $x \geq 0,$ $y \geq 0$ ; $P = 7 x + 10 y$ ; maximum $P = $ 60$ at $(0, 6)$ A shop makes bookshelves ( $x$ ) and desks ( $y$ ). Each bookshelf earns $15 and each desk $11. Each bookshelf takes 4 hours of sanding and 4 hours of painting; each desk takes 2 hours of sanding and 3 hours of painting. There are 12 hours of sanding and 14 hours of painting available. Write the constraints and the profit, then maximize it. $4 x + 2 y \leq 12,$ $4 x + 3 y \leq 14,$ $x \geq 0,$ $y \geq 0$ ; $P = 15 x + 11 y$ ; maximum $P = $ 52$ at $(2, 2)$ A shop makes tables ( $x$ ) and chairs ( $y$ ). Each table earns $15 and each chair $6. Each table takes 4 hours of carpentry and 1 hour of finishing; each chair takes 1 hour of carpentry and 1 hour of finishing. There are 40 hours of carpentry and 16 hours of finishing available. Write the constraints and the profit, then maximize it. $4 x + y \leq 40,$ $x + y \leq 16,$ $x \geq 0,$ $y \geq 0$ ; $P = 15 x + 6 y$ ; maximum $P = $ 168$ at $(8, 8)$ A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $11 and each necklace $5. Each bracelet takes 1 minute of beading and 4 minutes of packing; each necklace takes 1 minute of beading and 1 minute of packing. There are 11 minutes of beading and 26 minutes of packing available. Write the constraints and the profit, then maximize it. $x + y \leq 11,$ $4 x + y \leq 26,$ $x \geq 0,$ $y \geq 0$ ; $P = 11 x + 5 y$ ; maximum $P = $ 85$ at $(5, 6)$ A shop makes small boxes ( $x$ ) and large boxes ( $y$ ). Each small box earns $6 and each large box $14. Each small box takes 3 minutes of cutting and 4 minutes of folding; each large box takes 1 minute of cutting and 4 minutes of folding. There are 9 minutes of cutting and 20 minutes of folding available. Write the constraints and the profit, then maximize it. $3 x + y \leq 9,$ $4 x + 4 y \leq 20,$ $x \geq 0,$ $y \geq 0$ ; $P = 6 x + 14 y$ ; maximum $P = $ 70$ at $(0, 5)$ A shop makes cakes ( $x$ ) and pies ( $y$ ). Each cake earns $11 and each pie $13. Each cake takes 2 hours of oven time and 3 hours of decorating; each pie takes 2 hours of oven time and 1 hour of decorating. There are 24 hours of oven time and 20 hours of decorating available. Write the constraints and the profit, then maximize it. $2 x + 2 y \leq 24,$ $3 x + y \leq 20,$ $x \geq 0,$ $y \geq 0$ ; $P = 11 x + 13 y$ ; maximum $P = $ 156$ at $(0, 12)$ 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 or minimum and where it happens.

Maximize over a feasible region
Maximize $P = 6 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 16$ , $4 x + 2 y \leq 18$ $P = 31$ at $(4, 1)$
Maximize $P = 9 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 7$ , $4 x + y \leq 6$ $P = 23$ at $(1, 2)$
Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $4 x + 4 y \leq 16$ , $2 x + y \leq 7$ $P = 36$ at $(0, 4)$
Maximize $P = 8 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 2 y \leq 14$ , $4 x + 4 y \leq 36$ $P = 72$ at $(9, 0)$
Maximize $P = 2 x + 7 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 4 y \leq 20$ , $x + y \leq 6$ $P = 35$ at $(0, 5)$
Maximize $P = 9 x + 2 y$ subject to $x \geq 0$ , $y \geq 0$ , $x + 3 y \leq 13$ , $x + 4 y \leq 16$ $P = 117$ at $(13, 0)$
Maximize $P = 2 x + 8 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + 3 y \leq 12$ , $x + 2 y \leq 6$ $P = 24$ at $(0, 3)$

Minimize over a feasible region
Minimize $C = 2 x + y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 2 y \geq 18,$ $x + 2 y \geq 15$ $C = 9$ at $(0, 9)$
Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 4 y \geq 17,$ $4 x + 3 y \geq 18$ $C = 30$ at $(3, 2)$
Minimize $C = 2 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + y \geq 11,$ $3 x + 3 y \geq 21$ $C = 14$ at $(7, 0)$
Minimize $C = 5 x + 3 y$ subject to $x \geq 0,$ $y \geq 0,$ $x + 2 y \geq 18,$ $x + y \geq 12$ $C = 36$ at $(0, 12)$
Minimize $C = 6 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + 4 y \geq 18,$ $2 x + y \geq 6$ $C = 30$ at $(1, 4)$
Minimize $C = 4 x + 5 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + 2 y \geq 26,$ $x + 4 y \geq 22$ $C = 44$ at $(6, 4)$
Minimize $C = 4 x + 6 y$ subject to $x \geq 0,$ $y \geq 0,$ $2 x + y \geq 10,$ $x + 2 y \geq 8$ $C = 28$ at $(4, 2)$

Solve linear programming word problems
A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $7 and each necklace $10. Each bracelet takes 4 minutes of beading and 1 minute of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 16 minutes of beading and 6 minutes of packing available. Write the constraints and the profit, then maximize it. $4 x + 2 y \leq 16,$ $x + y \leq 6,$ $x \geq 0,$ $y \geq 0$ ; $P = 7 x + 10 y$ ; maximum $P = $ 60$ at $(0, 6)$
A shop makes bookshelves ( $x$ ) and desks ( $y$ ). Each bookshelf earns $15 and each desk $11. Each bookshelf takes 4 hours of sanding and 4 hours of painting; each desk takes 2 hours of sanding and 3 hours of painting. There are 12 hours of sanding and 14 hours of painting available. Write the constraints and the profit, then maximize it. $4 x + 2 y \leq 12,$ $4 x + 3 y \leq 14,$ $x \geq 0,$ $y \geq 0$ ; $P = 15 x + 11 y$ ; maximum $P = $ 52$ at $(2, 2)$
A shop makes tables ( $x$ ) and chairs ( $y$ ). Each table earns $15 and each chair $6. Each table takes 4 hours of carpentry and 1 hour of finishing; each chair takes 1 hour of carpentry and 1 hour of finishing. There are 40 hours of carpentry and 16 hours of finishing available. Write the constraints and the profit, then maximize it. $4 x + y \leq 40,$ $x + y \leq 16,$ $x \geq 0,$ $y \geq 0$ ; $P = 15 x + 6 y$ ; maximum $P = $ 168$ at $(8, 8)$
A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $11 and each necklace $5. Each bracelet takes 1 minute of beading and 4 minutes of packing; each necklace takes 1 minute of beading and 1 minute of packing. There are 11 minutes of beading and 26 minutes of packing available. Write the constraints and the profit, then maximize it. $x + y \leq 11,$ $4 x + y \leq 26,$ $x \geq 0,$ $y \geq 0$ ; $P = 11 x + 5 y$ ; maximum $P = $ 85$ at $(5, 6)$
A shop makes small boxes ( $x$ ) and large boxes ( $y$ ). Each small box earns $6 and each large box $14. Each small box takes 3 minutes of cutting and 4 minutes of folding; each large box takes 1 minute of cutting and 4 minutes of folding. There are 9 minutes of cutting and 20 minutes of folding available. Write the constraints and the profit, then maximize it. $3 x + y \leq 9,$ $4 x + 4 y \leq 20,$ $x \geq 0,$ $y \geq 0$ ; $P = 6 x + 14 y$ ; maximum $P = $ 70$ at $(0, 5)$
A shop makes cakes ( $x$ ) and pies ( $y$ ). Each cake earns $11 and each pie $13. Each cake takes 2 hours of oven time and 3 hours of decorating; each pie takes 2 hours of oven time and 1 hour of decorating. There are 24 hours of oven time and 20 hours of decorating available. Write the constraints and the profit, then maximize it. $2 x + 2 y \leq 24,$ $3 x + y \leq 20,$ $x \geq 0,$ $y \geq 0$ ; $P = 11 x + 13 y$ ; maximum $P = $ 156$ at $(0, 12)$

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.

What you can put on this worksheet

Maximize over a feasible region: Maximize $P = 3 x + 9 y$ subject to $x \geq 0$ , $y \geq 0$ , $3 x + y \leq 12$ , $2 x + 3 y \leq 15$ → $P = 45$ at $(0, 5)$
Minimize: Minimize $C = x + 4 y$ subject to $x \geq 0,$ $y \geq 0,$ $3 x + y \geq 16,$ $2 x + 3 y \geq 20$ → $C = 10$ at $(10, 0)$
From a word problem: A shop makes bracelets ( $x$ ) and necklaces ( $y$ ). Each bracelet earns $6 and each necklace $3. Each bracelet takes 1 minute of beading and 3 minutes of packing; each necklace takes 2 minutes of beading and 1 minute of packing. There are 19 minutes of beading and 22 minutes of packing available. Write the constraints and the profit, then maximize it. → $x + 2 y \leq 19,$ $3 x + y \leq 22,$ $x \geq 0,$ $y \geq 0$ ; $P = 6 x + 3 y$ ; maximum $P = $ 51$ at $(5, 7)$

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.

You pick from 3: maximize over a feasible region, minimize and from a word problem. Tick as many as you want and set how many of each, or let it spread them evenly.

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

Linear programming worksheet

Maximize over a feasible region

Minimize over a feasible region

Solve linear programming word problems

Maximize over a feasible region

Minimize over a feasible region

Solve linear programming word problems

How to do these

Where students go wrong

What you can put on this worksheet

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 kinds of question are on a linear programming worksheet?+

What do students get wrong with linear programming?+