Let's denote the cost of a shirt as \( S \) and the cost of a sweater as \( W \).
From the problem statement, we can set up the following equations based on the combinations of shirts and sweaters Andrea can buy:
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For 3 shirts and 2 sweaters: \[ 3S + 2W = 16 \]
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For 2 shirts and 4 sweaters: \[ 2S + 4W = 24 \]
Now, we solve these equations step by step.
First, we can simplify the second equation: \[ 2S + 4W = 24 \implies S + 2W = 12 \quad \text{(divide both sides by 2)} \]
Now we have the two equations:
- \( 3S + 2W = 16 \)
- \( S + 2W = 12 \)
Next, we can solve for \( S \) in terms of \( W \) using the second equation: \[ S = 12 - 2W \]
Now we'll substitute \( S \) into the first equation: \[ 3(12 - 2W) + 2W = 16 \] Expanding this gives: \[ 36 - 6W + 2W = 16 \] Combining like terms: \[ 36 - 4W = 16 \] Subtracting 36 from both sides: \[ -4W = 16 - 36 \implies -4W = -20 \] Dividing both sides by -4: \[ W = 5 \]
Now substitute \( W = 5 \) back into the equation \( S + 2W = 12 \): \[ S + 2(5) = 12 \] \[ S + 10 = 12 \] Subtracting 10 from both sides: \[ S = 2 \]
Thus, we have:
- The cost of a shirt (S) is \( 2 \).
- The cost of a sweater (W) is \( 5 \).
The item with the lowest value is the shirt, costing \( 2 \).
From the given options:
- 16
- 24
- 4
- 8
None of these options directly match the cost of the shirt or sweater. If we instead need to find the item with the lowest value from the provided options without a match in our derived values, we should pick the lowest number, which would be 4.
Thus, the answer is 4.