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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:
15x + 5y ≤ 1000
2x + y ≤ 50
Which combinations of books, x, and magazines, y, can the bookstore order?
(1 point)
50 books and 0 magazines
10 books and 30 magazines
40 books and 5 magazines
20 books and 15 magazines

1 answer

To determine which combinations of books \(x\) and magazines \(y\) the bookstore can order, we must check each given combination against the provided inequalities:

  1. Budget inequality: \(15x + 5y \leq 1000\)
  2. Space inequality: \(2x + y \leq 50\)

Now, we will evaluate each combination.

1. \(x = 50\), \(y = 0\)

  • Check the budget inequality: \[ 15(50) + 5(0) = 750 \leq 1000 \quad (\text{True}) \]
  • Check the space inequality: \[ 2(50) + 0 = 100 \leq 50 \quad (\text{False}) \] Conclusion: Not a valid combination.

2. \(x = 10\), \(y = 30\)

  • Check the budget inequality: \[ 15(10) + 5(30) = 150 + 150 = 300 \leq 1000 \quad (\text{True}) \]
  • Check the space inequality: \[ 2(10) + 30 = 20 + 30 = 50 \leq 50 \quad (\text{True}) \] Conclusion: Valid combination.

3. \(x = 40\), \(y = 5\)

  • Check the budget inequality: \[ 15(40) + 5(5) = 600 + 25 = 625 \leq 1000 \quad (\text{True}) \]
  • Check the space inequality: \[ 2(40) + 5 = 80 + 5 = 85 \leq 50 \quad (\text{False}) \] Conclusion: Not a valid combination.

4. \(x = 20\), \(y = 15\)

  • Check the budget inequality: \[ 15(20) + 5(15) = 300 + 75 = 375 \leq 1000 \quad (\text{True}) \]
  • Check the space inequality: \[ 2(20) + 15 = 40 + 15 = 55 \leq 50 \quad (\text{False}) \] Conclusion: Not a valid combination.

Summary

  • Valid combination: \(10\) books and \(30\) magazines.
  • Invalid combinations: \(50\) books and \(0\) magazines, \(40\) books and \(5\) magazines, \(20\) books and \(15\) magazines.

The only combination that meets both inequalities is 10 books and 30 magazines.