In the context of the situation involving Santiago purchasing books for the day care, the main constraints will arise from the nature of the problem, particularly regarding the quantities of the books he can buy. Here are the key constraints to consider:
1. Non-negativity Constraints:
Both the number of board books and soft cover books must be non-negative integers. This means:
- \( x \geq 0 \) (number of board books)
- \( y \geq 0 \) (number of soft cover books)
Santiago cannot buy a negative number of books, and since he is purchasing whole books, \( x \) and \( y \) must be integers.
2. Budget Constraint:
Santiago has a maximum budget of $20 for purchasing books. This is already encapsulated in the equation \( 1.50x + 0.50y = 20 \). This means that the total cost of the books cannot exceed $20, which is reflected in any combination he chooses for \( x \) and \( y \).
3. Practical Constraints (if applicable):
While not mathematically defined like the above constraints, there may be practical considerations:
- Availability of Books: There may not be an unlimited supply of board books or soft cover books at the sale. If the sale has limited quantities of each, this could constrain the maximum values of \( x \) and \( y \).
- Desired Book Types: Santiago might have preferences for certain types of books that could further limit his options.
Summary of Constraints:
The constraints for Santiago are summarized as follows:
- \( x \geq 0 \) (number of board books)
- \( y \geq 0 \) (number of soft cover books)
- \( 1.50x + 0.50y \leq 20 \) (total cost of books must not exceed $20)
These constraints will limit the feasible combinations of board books and soft cover books that Santiago can purchase for the day care.