Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ The members of a cooking club are making cakes, which they will sell at a street fair for $\$23$ apiece. It cost $\$20$ for a booth at the fair, and the ingredients for each cake cost $\$3$ . At some point, the club members will sell enough cakes so that their sales cover their expenditures. How many cakes will they have sold? How much will the sales and expenditures be? $\newline$ Once the members of the club have sold _ cakes, they will break even, with sales and expenditures of $\$$ _.

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Solve a system of equations using any method: word problems

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

\newline

The members of a cooking club are making cakes, which they will sell at a street fair for

\$23

apiece. It cost

\$20

for a booth at the fair, and the ingredients for each cake cost

\$3

. At some point, the club members will sell enough cakes so that their sales cover their expenditures. How many cakes will they have sold? How much will the sales and expenditures be?

\newline

Once the members of the club have sold _____ cakes, they will break even, with sales and expenditures of

\$

_____.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of cakes sold. $\newline$ Let $y$ be the total sales in dollars. $\newline$ We can write two equations to represent the situation: $\newline$ $1$ . The total sales ( $y$ ) will be equal to the number of cakes sold ( $x$ ) multiplied by the price per cake ( $\$23$ ). $\newline$ $2$ . The total expenditures will be the cost of the booth ( $\$20$ ) plus the cost of ingredients for each cake ( $\$3$ ) multiplied by the number of cakes sold ( $x$ ). $\newline$ The system of equations is: $\newline$ $y = 23x$ (Equation $1$ : Sales) $\newline$ $y = 20 + 3x$ (Equation $2$ : Expenditures) $\newline$ We want to find the point where sales equal expenditures, so we set the two equations equal to each other: $\newline$ $y$ $0$
Write Equations: Now, we solve for $x$ : $23x = 20 + 3x$ Subtract $3x$ from both sides to get: $20x = 20$
Set Equations Equal: Divide both sides by $20$ to find the number of cakes: $\newline$ $x = \frac{20}{20}$ $\newline$ $x = 1$
Solve for x: Now that we know the number of cakes sold $x$ , we can find the total sales $y$ by substituting $x$ back into either Equation $1$ or Equation $2$ . We'll use Equation $1$ : $\newline$ $y = 23x$ $\newline$ $y = 23(1)$ $\newline$ $y = \$(23)$
Find Total Sales: We have found that the members of the club will break even after selling $1$ cake, with sales and expenditures of $\$23$ .