Expert: Bobby Soltani - 1/13/2005

Question
Todd purchases all of his CDs in a local store.  All CDs in this store cost $13.99.  Todd's sister Shelli has just decided to join a mail-order music club.  Her first ten CDs are free.  After that, each CD will cost her $16.99(including shipping charges.)  Use this informationfor the activity below.

Write an equation that models the cost of Todd's CD purchases and the cost of Shelli's CD purchases.  Let y be the total cost of the CDs, and let be the number of CDs purchases.  Use the substitution method to solve this system of equations.  Veriry your answer by solving the system of equations using combination method.  Finally, graph the two equations.

1. When will Todd and Shelli have the same amount for CDs?  How many CDs will they have purchased?

2. When would it be less expensive to buy CDs from the music store?  When would it be less expensive to buy CDs through mail?

Answer
Hi Jennie,

First we will set up the equation for Todd. The cost of the CDs(y) is equal to 13.99 times the number of CDs (x) that he buys. So we can write,

y = 13.99x

For Shelly, we know that the first 10 CDs are free, and that they cost 16.99 after that. We can write this as

y = (x-10)*16.99, for x >= 10, and y = 0 for 0 <= x < 10

This means that the cost is 0 for the 0-10 CDS and 16.99 for each additional CD.

We can see from the equation that it is going to be cheaper for Shelli at first because she gets 10 free, but then as she buys a lot of CDs, it will become more expensive for her because she has to pay 16.99 for each. The first questions wants to know what is the x value when the cost (y) is equal. So we have,

y = 13.99x
y = (x-10)*16.99
set the two equations equal to each other
13.99x = (x-10)*16.99
multiply the 16.99 through the parenthesis.
13.99x = 16.99x - 169.90
subtract 13.99x from both sides
0 = 3x - 169.90
add 169.90 to both sides
169.90 = 3x
divide both sides by 3
x = 56.63

That means that they will have spent the same amount between cd number 56 and 57.

2. It would be less expensive to buy cds from the music store if you were buying 57 or more CDs. It would be less expensive to buy CDs through the mail if you were buying 56 or less CDs.

Let me know if you have specific questions.

Bobby