To formulate the two-variable inequality based on Kiani's budget and the costs of books, let's define the variables:
- Let \( x \) be the number of hardbacks purchased.
- Let \( y \) be the number of paperbacks purchased.
The costs can be represented as follows:
- The cost of hardbacks is \( 12x \).
- The cost of paperbacks is \( 5y \).
Since Kiani has a budget of $60, we can write the inequality representing this constraint as:
\[ 12x + 5y \leq 60 \]
Next, we need to determine the constraints of the problem:
- The number of hardbacks \( x \) must be non-negative: \( x \geq 0 \).
- The number of paperbacks \( y \) must also be non-negative: \( y \geq 0 \).
Thus, the complete set of constraints can be summarized as:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 12x + 5y \leq 60 \)
Now, let's evaluate the given points to see if they are viable solutions based on the constraints:
-
Point (−2, 6):
- \( x = -2 \) (not valid because \( x \) must be \( \geq 0 \))
- Not a viable solution.
-
Point (6, 6):
- \( x = 6 \) and \( y = 6 \)
- Check the budget: \( 12(6) + 5(6) = 72 + 30 = 102 \) (exceeds $60)
- Not a viable solution.
-
Point (2, 4):
- \( x = 2 \) and \( y = 4 \)
- Check the budget: \( 12(2) + 5(4) = 24 + 20 = 44 \) (within $60)
- This is a viable solution.
-
Point (1, 4.5):
- \( x = 1 \) and \( y = 4.5 \)
- Check the budget: \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \) (within $60)
- This is also a viable solution.
In conclusion, the points that represent viable solutions are (2, 4) and (1, 4.5). However, since you asked for one viable solution from the list, the answer would be:
(2, 4).
2 answers