Question

Write a system of equations to describe the situation below, solve using substitution, and fillin the blanks.Cooper is starting a business selling handmade necklaces. He has decided to invest an initialamount of $24 for advertising, and materials cost $8 for each necklace he makes. Cooper cansell his creations for $14 per necklace. Once he makes and sells a certain number ofnecklaces, he will break even, with identical expenses and sales. What would the totalexpenses and sales be then? How many necklaces would that take?

Answer 1

Given:

For expenses:

`\begin{gathered} Total\text{ expenses}=advertising\text{ fee}+material\text{ cost} \\ Let\text{ the number of necklaces be represented by }x \\ Total\text{ expenses}=24+8x \end{gathered}`

For sales:

`\begin{gathered} \text{ \$14 per necklace} \\ Total\text{ sales}=14x \end{gathered}`

At break-even, he will neither make profit nor loss.

At break-even, the expenses are equal to the sales.

Hence,

`\begin{gathered} 24+8x=14x \\ Collecting\text{ the like terms,} \\ 24=14x-8x \\ 24=6x \\ Dividing\text{ both sides by 6 to get }x; \\ (24)/(6)=x \\ x=4\text{ necklaces} \end{gathered}`

Hence,

`\begin{gathered} Total\text{ expenses}=24+8x \\ Total\text{ expenses}=24+8(4) \\ Total\text{ expenses}=24+32 \\ Total\text{ expenses}=\text{ \$56} \\ \\ \\ Total\text{ sales}=14x \\ Total\text{ sales}=14(4) \\ Total\text{ sales}=\text{ \$}56 \end{gathered}`

Therefore, at break-even, the total expenses will be $56 and the total sales will be $56.

It will take the sales of 4 necklaces to reach break-even.