Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Cole's Bakery recently spent a total of $\$299$ on new equipment, and their average hourly operating costs are $\$13$ . Their average hourly receipts are $\$36$ . The bakery will soon make back the amount it invested in equipment. How many hours will that take? What would the total expenses and receipts both equal? $\newline$ In $\_\_\_\_$ hours, the bakery's total expenses and receipts will both equal $\$\_\_\_\_\_$ .

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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

Cole's Bakery recently spent a total of

\$299

on new equipment, and their average hourly operating costs are

\$13

. Their average hourly receipts are

\$36

. The bakery will soon make back the amount it invested in equipment. How many hours will that take? What would the total expenses and receipts both equal?

\newline

\_\_\_\_

hours, the bakery's total expenses and receipts will both equal

\$\_\_\_\_\_

Define Variables: Let's define two variables: let $x$ be the number of hours, and let $y$ be the total amount of money made back. We can write two equations to represent the situation. The first equation will represent the total expenses, and the second equation will represent the total receipts.
Calculate Total Expenses: The total expenses can be calculated by multiplying the average hourly operating costs by the number of hours, plus the initial investment in equipment. This gives us the equation: $\newline$ $y = 13x + 299$
Calculate Total Receipts: The total receipts can be calculated by multiplying the average hourly receipts by the number of hours. This gives us the equation: $\newline$ $y = 36x$
Solve System of Equations: Now we have a system of equations: $\newline$ $1$ ) $y = 13x + 299$ $\newline$ $2$ ) $y = 36x$ $\newline$ We can solve this system using substitution by setting the two expressions for $y$ equal to each other.
Substitute and Simplify: Setting the two expressions for $y$ equal to each other, we get: $13x + 299 = 36x$
Solve for x: To solve for x, we need to get all the x terms on one side. We can do this by subtracting $13x$ from both sides of the equation: $\newline$ $13x + 299 - 13x = 36x - 13x$ $\newline$ This simplifies to: $\newline$ $299 = 23x$
Calculate $x$ : Now, we divide both sides by $23$ to solve for $x$ : $\frac{299}{23} = x$
Find Total Expenses and Receipts: Calculating the division gives us: $\newline$ $x \approx 13$ $\newline$ Since the number of hours cannot be a fraction, we round up to the nearest whole number if necessary. However, in this case, $299$ is divisible by $23$ , so $x$ is exactly $13$ .
Calculate Final Total: Now that we know $x = 13$ , we can find the total expenses and receipts by substituting $x$ back into either of the original equations. We'll use the receipts equation for simplicity: $\newline$ $y = 36x$ $\newline$ $y = 36(13)$
Calculate Final Total: Now that we know $x = 13$ , we can find the total expenses and receipts by substituting $x$ back into either of the original equations. We'll use the receipts equation for simplicity: $\newline$ $y = 36x$ $\newline$ $y = 36(13)$ Calculating the multiplication gives us: $\newline$ $y = 468$ $\newline$ So, the total expenses and receipts will both equal $\$468$ after $13$ hours.