To set up the inequalities that represent the constraints on the system based on the given situation, we need to consider both the total expenditure and the total number of items.
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Total Cost Constraint: The coach spends a total of $140 on drinks and food items. Each drink costs $5 (x) and each food item costs $8 (y). Therefore, we can write the inequality for the total cost as: \[ 5x + 8y \leq 140 \]
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Total Items Constraint: The coach buys a total of 22 drinks and food items, which can be expressed as: \[ x + y = 22 \]
Next, the variables x (number of drinks) and y (number of food items) must also be non-negative: \[ x \geq 0 \ y \geq 0 \]
Now, putting the constraints together, we have:
For the constraint on the number of drinks: \[ 0 \leq x \leq 22 \]
For the constraint on the number of food items: \[ 0 \leq y \leq 22 \]
However, since we know from the total items constraint that \(x + y = 22\), these inequalities will really limit \(y\) based on the value of \(x\). Specifically, if \(x\) goes up to 22, then \(y\) can go down to 0, since together they must equal 22. Similarly, if \(x\) is 0, then \(y\) would be 22.
So, we can summarize the inequalities for \(x\) and \(y\) in the context of the specific constraints for this scenario as follows:
Summary of inequalities:
- For drinks: \(0 \leq x \leq 22\)
- For food items: \(0 \leq y \leq 22\)
Please confirm if you would like additional elaboration on any part of this explanation or further breakdown of these inequalities.