To determine if a point is a solution in the context of Meha's birthday party planning problem, let's first establish the situation mathematically.
Let:
- \( c \) = number of children
- \( a \) = number of adults
The total cost for the party can be represented by the equation: \[ \text{Total Cost} = 150 + 10.50c + 5.50a \] We know that the maximum budget is $300, so we can write the inequality: \[ 150 + 10.50c + 5.50a \leq 300 \] This simplifies to: \[ 10.50c + 5.50a \leq 150 \]
Additionally, since there needs to be at least one child and one adult, we also have the constraints: \[ c \geq 1 \quad \text{and} \quad a \geq 1 \]
Now, you need to check if a particular point (which should be something like \( (c, a) \)) satisfies the inequality \( 10.50c + 5.50a \leq 150 \) and meets the constraints \( c \geq 1 \) and \( a \geq 1 \).
Steps for the problem:
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If you provide a point \( (c, a) \), we can substitute those values into our inequalities and determine if they satisfy both the cost limit and the minimum requirements for both children and adults.
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If, for example, the point given is \( (5, 5) \):
- Check the cost: \( 10.50(5) + 5.50(5) = 52.50 + 27.50 = 80 \)
- Total cost = \( 150 + 80 = 230 \leq 300 \) (this part is satisfied)
- Check minimum constraints: \( c = 5 \geq 1 \) and \( a = 5 \geq 1 \) (this part is also satisfied)
Given this analysis structure, please provide the specific point (i.e., values for \( c \) and \( a \)), and I'll tell you whether it's a solution in the context of Meha's party planning!
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