Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for $114

Question

, or she is able to buy 2 shirts and 4 sweaters for $164
. How much does the item with the lowest value cost?

16

24

4

8

Answer 1

Let the cost of a shirt be \( S \) and the cost of a sweater be \( W \). We can set up the following equations based on the information given:

1. \( 3S + 2W = 114 \) (from buying 3 shirts and 2 sweaters)
2. \( 2S + 4W = 164 \) (from buying 2 shirts and 4 sweaters)

We can simplify the second equation. Dividing everything by 2 gives us:

\[ S + 2W = 82 \]

Now we have the system of equations:

1. \( 3S + 2W = 114 \)
2. \( S + 2W = 82 \)

Next, we can eliminate \( W \) by subtracting the second equation from the first:

\[ (3S + 2W) - (S + 2W) = 114 - 82 \]

This simplifies to:

\[ 2S = 32 \] \[ S = 16 \]

Now that we have the price of a shirt \( S \), we can substitute \( S \) back into the second equation to find \( W \):

\[ 16 + 2W = 82 \] \[ 2W = 82 - 16 \] \[ 2W = 66 \] \[ W = 33 \]

Thus, the cost of a shirt is \( 16 \) and the cost of a sweater is \( 33 \).

The item with the lowest value is the shirt, which costs \( \boxed{16} \).