Question

company a sells hats for \\$11 per hat plus a fixed \\$7 shipping and handling fee. company b sells hats for \\$8 per hat plus a fixed \\$16 shipping and handling fee. the marketing director of company a wants to set up a system of equations to find the number of hats for which the total cost of an order for the two companies is the same. he uses the variables \\(x\\) and \\(y\\). he lets \\(y\\) represent the total cost of an order. what will \\(x\\) represent?

\\(x = \text{total cost of an order}\\)
\\(x = \text{shipping costs for company b}\\)
\\(x = \text{shipping costs for company a}\\)
\\(x = \text{number of hats}\\)

Explanation:

Identify the given information

Using the Variable Identification knowledge point

• Company A: \$11 per hat plus \$7 shipping.
• Company B: \$8 per hat plus \$16 shipping.
• \(y\) represents the total cost of an order.

Determine the role of the independent variable

Using the System of Equations Translation knowledge point

• The total cost \(y\) depends on the number of hats purchased.
• The rate is given per hat (\$11/hat and \$8/hat).
• Therefore, the variable \(x\) must represent the number of hats.

Match with the given options

Using the Variable Identification knowledge point

• Option 1: \(x = \text{total cost of an order}\) (Incorrect, this is \(y\)).
• Option 2: \(x = \text{shipping costs for company B}\) (Incorrect, this is a fixed \$16).
• Option 3: \(x = \text{shipping costs for company A}\) (Incorrect, this is a fixed \$7).
• Option 4: \(x = \text{number of hats}\) (Correct).

Answer:

• \(x = \text{total cost of an order}\)
• \(x = \text{shipping costs for company B}\)
• \(x = \text{shipping costs for company A}\)
• \(x = \text{number of hats}\) (Correct answer)