# The Algebraic Method

Select one of the real-world situations listed below. These situations require a mathematical comparison of cost options in order to determine which offers the greatest economic advantage.

• Compare two monthly phone calling plans to determine which plan is most advantageous based upon a customer’s monthly calling needs.

• Determine whether an outfitter leading a backpacking trip would find it more cost effective to make a one-time bulk foods purchase at a buyer’s club with membership fees or purchase the same goods at a conventional supermarket.

• Compare a yearly pass with a daily pass at a health club, zoo, museum, or theme park to determine which is most advantageous based upon the number of yearly visits.

• A teacher is looking for the best option in purchasing school supplies for a classroom. Company A is offering a discount for every dollar amount spent; Company B is offering a higher discount than Company A, but the discount is applicable only for every dollar amount spent above \$20. Determine which company will offer a better price based upon the dollar amount the teacher spends on school supplies.

Compare a yearly pass with a daily pass at a health club swimming pool. Yearly price is 120.00 a year for the health club swimming pool. Daily pass is 8.00

A. Introduce the problem by doing the following:

1. Describe the selected real-world scenario.

2. Explain the needs of the hypothetical customer.

3. Discuss two cost options that are being considered.

B. Evaluate the cost of each option algebraically by doing the following:

1. Develop a linear algebraic equation with clearly defined variables to represent the cost of each option.

2. Explain the reasoning behind the way each equation is constructed.

3. Solve the system of linear equations algebraically using either the substitution method or the elimination method to determine where the two cost options are equivalent, showing all work.

a. Explain each step used to solve the system of linear equations. Include the following in your explanation:

• The algebraic method used to solve the system of equations

• All mathematical operations used to solve the system of equations

• The solution of the system of equations

C. Depict the situation graphically, using an appropriate graphing software such as Excel or an online graphing utility. Include the following in your graph:

• A depiction of each cost option on the same graph

• Clear labels of the cost to the hypothetical customer for both options on the graph

• The intersection of the cost options (if one exists)

D. Discuss your recommendation to the customer based on both mathematical reasoning and contextual details. Your discussion should include the following:

• How the algebraic evaluation and graph support the recommendation

• Any relevant contextual details that pertain to the recommendation (e.g., flexibility, distance, availability, etc.) 1. Discuss the different options and the customer’s needs. Discussion (using graph)

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