Lesson Worksheet: Linear Programming Mathematics • 9th Grade

In this worksheet, we will practice finding the optimal solution of a linear system that has an objective function and multiple constraints.


Determine the values of π‘₯ and 𝑦 that maximize the function 𝑝=5π‘₯+2𝑦. Write your answer as a point (π‘₯,𝑦).

  • A(0,8)
  • B(7,8)
  • C(7,0)
  • D(3,0)


Using linear programming, find the minimum and maximum values of the function 𝑝=4π‘₯βˆ’3𝑦 given that π‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦≀9, and 𝑦β‰₯5 .

  • AThe minimum value is βˆ’27, and the maximum value is 1.
  • BThe minimum value is βˆ’15, and the maximum value is 1.
  • CThe minimum value is βˆ’27, and the maximum value is βˆ’15.
  • DThe minimum value is 0, and the maximum value is 9.


A seafood restaurant sells two types of cooked fish; cod and eel. The restaurant sells NO LESS than 40 fish every day but it does not use more than 30 cod and no more than 45 eels. The price of one cod is 6 LE and that of an eel is 8 LE. Let π‘₯ represent the amount of cod purchased each day, and 𝑦 represent the amount of eel. Given that the manager wants to minimise the total price, 𝑝, of fish, state the objective function and the inequalities that will help the restaurant manager decide how many of each fish to buy.

  • Aπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯40, π‘₯<30, 𝑦<45, 𝑝=6π‘₯+8𝑦
  • Bπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦>40, π‘₯≀30, 𝑦≀45, 𝑝=6π‘₯+8𝑦
  • Cπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯40, π‘₯β‰₯30, 𝑦β‰₯45, 𝑝=6π‘₯+8𝑦
  • Dπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯40, π‘₯≀30, 𝑦≀45, 𝑝>6π‘₯+8𝑦
  • Eπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯40, π‘₯≀30, 𝑦≀45, 𝑝=6π‘₯+8𝑦


A candy store sells bags of marshmallows for 5 LE each and bags of cola candy for 6 LE each. A child wants to buy both types of candy and has restrictions on how many they can buy that are described by the figure shown, where π‘₯ represents the number of bags of marshmallows they buy and 𝑦 represents the number of bags of cola candy. What is the lowest price possible in this situation?


Complete the following: The linear inequalities or restrictions on the variables of a linear programming problem are called .

  • Aminimum values
  • Bobjective functions
  • Cconstraints
  • Dmaximum values
  • Eoptimal values


Consider the following inequalities in the nonnegative variables π‘₯, π‘₯, and π‘₯: π‘₯βˆ’π‘₯+π‘₯≀10,π‘₯+π‘₯+π‘₯β‰₯1,π‘₯+2π‘₯+π‘₯≀7. Find the maximum and minimum possible values of 𝑧=2π‘₯+π‘₯ subject to these constraints.

  • Aminimum: 1, maximum: 14
  • Bminimum: 1, maximum: 7
  • Cminimum: 0, maximum: 14
  • Dminimum: 0, maximum: 7


Find the maximum value of the objective function 𝑝=2π‘₯+6𝑦 given the constraints π‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦≀6, 3π‘₯+𝑦≀9, and π‘₯+2𝑦≀8.


In a workshop, two workers produce two types of iron desks: type A and type B. One worker builds the desks and the other sprays them. It takes the first worker 4 hours to build one desk of type A and 3 hours to build one desk of type B. It takes the second worker 3 hours to spray one desk of type A and 4 hours to spray one desk of type B. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of 60 LE from each desk (of either type), determine the objective function and inequalities required for calculating the number of desks of each type to be produced every day to maximize the profit 𝑝.

  • Aπ‘₯β‰₯0, 𝑦β‰₯0, 4π‘₯+3𝑦>5, 3π‘₯+4𝑦<7, 𝑝β‰₯60π‘₯+60𝑦
  • Bπ‘₯β‰₯0, 𝑦β‰₯0, 4π‘₯+3𝑦≀5, 3π‘₯+4𝑦β‰₯7, 𝑝=60π‘₯+60𝑦
  • Cπ‘₯β‰₯0, 𝑦β‰₯0, 4π‘₯+3𝑦β‰₯5, 3π‘₯+4𝑦≀7, 𝑝=60π‘₯+60𝑦
  • Dπ‘₯β‰₯0, 𝑦β‰₯0, 4π‘₯+3𝑦β‰₯5, 3π‘₯+4𝑦β‰₯7, 𝑝β‰₯60π‘₯+60𝑦
  • Eπ‘₯β‰₯0, 𝑦β‰₯0, 4π‘₯+3𝑦<5, 3π‘₯+4𝑦>7, 𝑝=60π‘₯+60𝑦


A farmer can improve the quality of his produce if he uses at least 18 units of nitrogen-based compounds and at least 6 units of phosphate compounds. He can use two types of fertilizers: A and B. The cost and contents of each fertilizer are shown in the table.

The FertilizerNumber of Units of Nitrogen-Based Compounds per KilogramNumber of Units of Phosphate Compounds per KilogramCost for Each Kilogram (LE)
A 3 2 170
B 6 1 120

Given that the graph represents the constraints in this situation, find the lowest cost the farmer can pay for fertilizer while providing sufficient amounts of both compounds.


A small company dyes shirts to be either solid color or tie-dye and they want to decide how many shirts of each color to prepare for an upcoming sale. The company has a budget of $240. Purchasing an undyed shirt costs $2. It costs an additional $0.50 to dye a shirt with a solid color and $1.50 to dye a shirt with a tie-dye pattern. The company only has 8 hours to prepare all the shirts for the sale, and it takes 2 minutes to produce a solid-color shirt and 10 minutes to produce a tie-dye shirt. They decide to make a graph to help them maximise profit, given that they make $8 profit for each solid-color shirt and $10 for each tie-dye shirt.

Let π‘₯ represent the number of solid-color shirts and 𝑦 represent the number of tie-dye shirts. Which of the following shows the feasible region?

  • A
  • B
  • C
  • D

State the objective function.

  • A𝑓(π‘₯,𝑦)=2π‘₯+1.5𝑦+240
  • B𝑓(π‘₯,𝑦)=2π‘₯+10π‘¦βˆ’480
  • C𝑓(π‘₯,𝑦)=8π‘₯+10𝑦
  • D𝑓(π‘₯,𝑦)=2.5π‘₯+3.5π‘¦βˆ’240
  • E𝑓(π‘₯,𝑦)=2π‘₯+1.5𝑦

How many of each type of shirt should the company produce to maximize profit?

  • A0 solid-color shirts and 48 tie-dye shirts
  • B89 solid-color shirts and 69 tie-dye shirts
  • C40 solid-color shirts and 40 tie-dye shirts
  • D96 solid-color shirts and 0 tie-dye shirts
  • E48 solid-color shirts and 0 tie-dye shirts

This lesson includes 20 additional questions and 162 additional question variations for subscribers.

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