A bookstore is placing an order for new books and magazines. Each book costs $15, and each magazine costs $5. The store can spend at most $1,000 on these books and magazines. Additionally, each book takes up 2 square feet of space, and each magazine takes up 1 square foot of space. The store has only 50 square feet of display space available for the new items. The following system of inequalities can be used to determine possible combinations of books, x, and magazines, y, the store can order:

Question

15x+5y≤1000


2x+y≤50


Which combinations of books, x, and magazines, y, can the bookstore order?

(1 point)
Responses

20 books and 15 magazines
20 books and 15 magazines

50 books and 0 magazines
50 books and 0 magazines

10 books and 30 magazines
10 books and 30 magazines

40 books and 5 magazines

Answer 1

To determine which combinations of books (x) and magazines (y) the bookstore can order, we need to evaluate each option against the given system of inequalities:

1. Budget Constraint:
   \( 15x + 5y \leq 1000 \)

2. Space Constraint:
   \( 2x + y \leq 50 \)

Let's evaluate each combination:

Option 1: 20 books and 15 magazines (x = 20, y = 15)

• For the budget:
  \( 15(20) + 5(15) = 300 + 75 = 375 \)
  \( 375 \leq 1000 \) (satisfied)

• For the space:
  \( 2(20) + 15 = 40 + 15 = 55 \)
  \( 55 \leq 50 \) (not satisfied)

Option 2: 50 books and 0 magazines (x = 50, y = 0)

• For the budget:
  \( 15(50) + 5(0) = 750 + 0 = 750 \)
  \( 750 \leq 1000 \) (satisfied)

• For the space:
  \( 2(50) + 0 = 100 + 0 = 100 \)
  \( 100 \leq 50 \) (not satisfied)

Option 3: 10 books and 30 magazines (x = 10, y = 30)

• For the budget:
  \( 15(10) + 5(30) = 150 + 150 = 300 \)
  \( 300 \leq 1000 \) (satisfied)

• For the space:
  \( 2(10) + 30 = 20 + 30 = 50 \)
  \( 50 \leq 50 \) (satisfied)

Option 4: 40 books and 5 magazines (x = 40, y = 5)

• For the budget:
  \( 15(40) + 5(5) = 600 + 25 = 625 \)
  \( 625 \leq 1000 \) (satisfied)

• For the space:
  \( 2(40) + 5 = 80 + 5 = 85 \)
  \( 85 \leq 50 \) (not satisfied)

Summary of Results

• Option 1: Not valid (space)
• Option 2: Not valid (space)
• Option 3: Valid (both constraints satisfied)
• Option 4: Not valid (space)

The only valid combination is:

10 books and 30 magazines.