Question

the expression \\(8x + 2(3y)\\) represents the total amount, in dollars, that davellyn spent on tank tops and scarves, where \\(x\\) represents the number of tank tops she bought and \\(y\\) represents the number of scarves she bought. which statement must be true about the amount davellyn spent?

she spent \\(8x\\) dollars on scarves.
she spent 8 dollars for each scarf.
she spent \\(6y\\) dollars on scarves.
she spent a total of 14 dollars.

Explanation:

Analyze the given expression

Using the Algebraic Modeling knowledge point

$$ E = 8x + 2(3y) $$

where:

• \(x\) represents the number of tank tops bought.
• \(y\) represents the number of scarves bought.
• The total expression represents the total amount spent in dollars.

Simplify the expression

We simplify the second term of the expression to find the total amount spent on scarves:

$$ 2(3y) = 6y $$

Thus, the expression can be rewritten as:

$$ 8x + 6y $$

Interpret each term

In the simplified expression \(8x + 6y\):

• The term \(8x\) represents the total amount spent on tank tops, meaning each tank top costs \$8.
• The term \(6y\) represents the total amount spent on scarves, meaning each scarf costs \$6.

Evaluate the given options

• She spent \(8x\) dollars on scarves: Incorrect, \(8x\) is the amount spent on tank tops.
• She spent 8 dollars for each scarf: Incorrect, each scarf costs 6 dollars.
• She spent \(6y\) dollars on scarves: Correct, as \(2(3y) = 6y\) represents the total cost for \(y\) scarves.
• She spent a total of 14 dollars: Incorrect, the total spent depends on the variables \(x\) and \(y\).

Answer:

• She spent \(8x\) dollars on scarves.
• She spent 8 dollars for each scarf.
• She spent \(6y\) dollars on scarves. (Correct answer)
• She spent a total of 14 dollars.