A vendor at a craft show sold items for $4.50, $6.00, and $7.50. Altogether, the vendor sold 87 items for a total of $489. The vendor sold 5 more items for $6.00 than for $7.50. Which system of equations could you use to determine how many of each item were sold?

Question

x + y + z = 489

z = y + 5

4.5x + 6y + 7.5z = 87

x + y + z = 87

z = y + 5

4.5x + 6y + 7.5z = 489

x + y + z = 489

y = z + 5

4.5x + 6y + 7.5z = 87

x + y + z = 87

y = z + 5

4.5x + 6y + 7.5z = 489

Answer 1

To set up the system of equations correctly based on the problem statement, we need to define the following variables:

• Let \( x \) be the number of items sold for $4.50.
• Let \( y \) be the number of items sold for $6.00.
• Let \( z \) be the number of items sold for $7.50.

Based on the information given:

1. Total items sold: The vendor sold a total of 87 items, which can be expressed as: \[ x + y + z = 87 \]

2. Total revenue: The total revenue from all sold items is $489, which can be expressed as: \[ 4.5x + 6y + 7.5z = 489 \]

3. Relationship between y and z: The vendor sold 5 more items for $6.00 than for $7.50, which can be expressed as: \[ y = z + 5 \quad \text{(or equivalently)} \quad z = y - 5 \]

Putting this all together, the correct system of equations is:

1. \( x + y + z = 87 \)
2. \( 4.5x + 6y + 7.5z = 489 \)
3. \( y = z + 5 \)

Therefore, the appropriate choice from the options provided is: \[ x + y + z = 87 \] \[ 4.5x + 6y + 7.5z = 489 \] \[ y = z + 5 \]