Question

Tom has a collection of 30 CDs and Nita has a collection of 18 CDs. Tom is adding 1 CD a month to his collection while Nita is adding 5 CDs a month to her collection. Write a system to find the number of months after which they will have the same number of CDs. Let x represent the number of months and y the number of CDs.

Answer 1

x = 3

Explanation:

We can create a system of equations to represent the situation we have been given.

Creating the Equations

In this system, we can create an equation to represent both people's collections.

First, we can create Nita's.

• To create the equation we can start with the constant. We are told she starts with 18 CDs. So, this will be one of the terms.
• Next, we can multiply x by 5. We can do this because x represents the number of months that have passed and 5 is the number of CDs she adds per month. This means 5x will be another term.
• Finally, the constant and variable will add up to the total number of CDs, which is represented by y. So, the expression will be set equal to y.

All of this creates the equation: 5x + 18 = y.

Next, we can do the same to find Tom's equation.

• Tom starts out with 30 CDs. So this will be the constant term.
• Each month he adds 1 CD. This means that we can multiply 1 by x to make the term 1x.
• Finally, like with Nita's this expression will be set equal to y.

This creates the second equation: 1x + 30 = y.

The system is:

5x + 18 = y

1x + 30 = y

Solving the System of Equations

One method to solve a system of equations is substitution. Substitution is when you substitute an expression for a variable. Since both expressions are set equal to y, we can substitute the expressions for y and set them equal to each other.

This creates the new equation: 5x + 18 = 1x + 30. Then, solve normally.

• First, subtract 1x from both sides

4x + 18 = 30

• Next, subtract 18 from both sides.

4x = 12

• Finally, divide both sides by 4

x = 3

This means after 3 months the number CDs will be equal.

Answer 2

3 months

Explanation:

Let x be the number of months CDs are added at Tom and Nita's respective rates of 1 and 5 CDs/month.

Tom's collection after x months is = 30 + 1x

Nita's collection after x months is = 18 + 5x

At some point they will have equal numbers of CD's, so set the equations equal to each other:

30 + 1x = 18 + 5x

4x = 12

x = 3 months

At 3 months both Tom and Nita will have the same number of CDs.

Tom: 30 + 1*3 = 33 CDs

Nita: 18 + 5*3 = 33 CDs