Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Zack and Eddie, both teachers, are adding books to their class libraries. Zack's classroom started out with a collection of only 20 books, but he plans to purchase an additional 2 books per week. Eddie's library started out with 9 books, and he has enough money in his budget to purchase another 3 books per week. At some point, the two teachers' libraries will contain the same number of books. How many books will each class have? How many weeks will that take?


The two teachers' libraries will each have in how many books in how many weeks.

Answer 1

Let's define the variables to represent the situation:

- Let \( x \) be the number of weeks that pass.
- Let \( Z \) be the number of books in Zack's library after \( x \) weeks.
- Let \( E \) be the number of books in Eddie's library after \( x \) weeks.

We can set up the equations based on the information provided:

1. **Zack's books**: He starts with 20 books and adds 2 each week. Therefore, the equation for Zack is:
\[
Z = 20 + 2x
\]

2. **Eddie's books**: He starts with 9 books and adds 3 each week. Therefore, the equation for Eddie is:
\[
E = 9 + 3x
\]

To find out when their libraries will contain the same number of books, we set the two equations equal to each other:
\[
20 + 2x = 9 + 3x
\]

Now, we can solve for \( x \):
1. Subtract \( 2x \) from both sides:
\[
20 = 9 + x
\]
2. Subtract 9 from both sides:
\[
20 - 9 = x
\]
\[
11 = x
\]

So, it will take 11 weeks for the two libraries to have the same number of books. Now we can find out how many books each library will have at that time by substituting \( x = 11 \) back into either equation (we can use either, but let's use Zack's):

Calculating for Zack:
\[
Z = 20 + 2(11) = 20 + 22 = 42
\]

Calculating for Eddie:
\[
E = 9 + 3(11) = 9 + 33 = 42
\]

Both libraries will have 42 books.

Thus, we conclude:
The two teachers' libraries will each have **42 books** in **11 weeks**.