Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Paula is starting a business selling handmade necklaces. She has decided to invest an initial amount of $\$41$ for advertising, and materials cost $\$8$ for each necklace she makes. Paula can sell her creations for $\$9$ per necklace. Once she makes and sells a certain number of necklaces, she will break even, with identical expenses and sales. What would the total expenses and sales be then? How many necklaces would that take? $\newline$ Paula would have total expenses and sales $\$$ _ after selling _ necklaces.

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Paula is starting a business selling handmade necklaces. She has decided to invest an initial amount of

\$41

for advertising, and materials cost

\$8

for each necklace she makes. Paula can sell her creations for

\$9

per necklace. Once she makes and sells a certain number of necklaces, she will break even, with identical expenses and sales. What would the total expenses and sales be then? How many necklaces would that take?

\newline

Paula would have total expenses and sales

\$

_____ after selling _____ necklaces.

Define Variables and Equations: Let's define the variables and write the equations. Let $x$ be the number of necklaces Paula makes and sells. Her total expenses will be the sum of the initial investment and the cost of materials for each necklace. Her total sales will be the price per necklace times the number of necklaces sold.
Set Break-Even Point: To find the break-even point, set the total expenses equal to total sales.
Solve for x: Solve for $x$ using simple algebra. Subtract $8x$ from both sides to isolate $x$ on one side of the equation.
Calculate Expenses and Sales: Calculate the total expenses and sales at the break-even point by substituting $x = 41$ back into either the expense or sales equation.