To formulate the problem, let \( x \) represent the number of hardbacks Kiani buys, and let \( y \) represent the number of paperbacks. The following conditions must be met based on Kiani's budget:
- The cost of the hardbacks is \( 12x \) (since each hardback costs $12).
- The cost of the paperbacks is \( 5y \) (assuming "SS" means $5 for each paperback).
Kiani's total cost must not exceed $80, which gives us the following inequality:
\[ 12x + 5y \leq 80 \]
Additionally, since Kiani cannot buy a negative number of books, we have the following constraints:
\[ x \geq 0 \ y \geq 0 \]
Now we have a system of inequalities that describe the constraints:
- \( 12x + 5y \leq 80 \)
- \( x \geq 0 \)
- \( y \geq 0 \)
Next, we must evaluate the proposed points to determine which one represents a viable solution (i.e., a point that satisfies all inequalities):
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\( (-2, 6) \): This point is invalid because \( x \) (the number of hardbacks) cannot be negative.
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\( (2, 4) \): Check:
- \( 12(2) + 5(4) = 24 + 20 = 44 \) (which is \( \leq 80 \), valid)
- \( 2 \geq 0 \) (valid)
- \( 4 \geq 0 \) (valid)
This point is a viable solution.
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\( (1, 4.5) \): Check:
- \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \) (which is \( \leq 80 \), valid)
- \( 1 \geq 0 \) (valid)
- \( 4.5 \geq 0 \) (valid)
This point is also a viable solution.
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\( (6, 6) \): Check:
- \( 12(6) + 5(6) = 72 + 30 = 102 \) (which is \( > 80 \), invalid)
- \( 6 \geq 0 \) (valid)
- \( 6 \geq 0 \) (valid)
This point is not a viable solution.
Thus, the viable solutions are \( (2, 4) \) and \( (1, 4.5) \).
In the choices given, the point that represents a viable solution is:
- (2, 4)
2 answers