Actually Applicable Application Problems and Brainteasers/Two-Variable Linear Programming

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This strategy uses graphing two-variable linear inequalities, solving systems of two linear equations, and evaluating formulas to make decisions between two options in business settings. It is a real application problem for any businessperson who needs to choose between two options that are related through linear constraints. There are also related strategies for dealing with situations with more than two variables.

General Method

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  1. Set up your objective function, z=ax+by, where a and b are usually the profit or revenue per item sold for items x and y.
  2. For each resource which is in limited supply, known as constraints (materials, time in a production facility, storage space, etc.), set up an inequality showing that the total amount used of that resource must be less than or equal to the maximum.
    • The inequalities you set up will often also be called constraints in discussing these types of problems.
  3. Graph all the inequalities and clearly shade the region where they overlap, called the feasible region.
  4. Use strategies for solving simultaneous equations to determine the coordinates of the corners of the feasible region.
  5. Evaluate the objective function at each corner to determine where its value is greatest.


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Xylophones and Yo-yos

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Situation: You are helping run a business which makes xylophones and premium yo-yos. You need to decide how many of each to make. The following list is a summary of the pertinent facts.

  • A xylophone sells for $12 and a yo-yo sells for $17.
  • A xylophone requires $3 in materials and a yo-yo requires $5.
  • You have $175 to work with to buy materials. (You have budgeted separately for the cost of your production facility, employees, etc.)
  • A xylophone takes 1/2 hour in the metal cutting machine and 2 hours in the polishing machine.
  • A yo-yo takes 3 hours in the metal cutting machine and 1/2 hour in the polishing machine.
  • You run your machines for two shifts, each of which is 8 hours, 5 days a week.
  • There is plenty of demand for both xylophones and yo-yos so you will sell whatever you make.

Goal: As in most businesses, your goal is to make as much profit as possible.

Make Your Own Problem

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Find a business with two products (or pick two products from a business which makes more) and research their constraints such as storage space, availability of materials, and tools and labor used in producing the two products. Also find out their profit on each of the two products to set up the objective function (or, if they won't give out that information, use their selling prices, and optimize for revenue instead or profit).