Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. 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? In _____ hours, the bakery's total expenses and receipts will both equal $_____.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  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?   In _____ hours, the bakery's total expenses and receipts will both equal $_____.

Bytelearn · Accepted Answer

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: y = 13x + 299Calculate Total Receipts: The total receipts can be calculated by multiplying the average hourly receipts by the number of hours. This gives us the equation: y = 36xSolve System of Equations: Now we have a system of equations: 1) y = 13x + 299 2) y = 36x 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 = 36xSolve 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: 13x + 299 - 13x = 36x - 13x This simplifies to: 299 = 23xCalculate x: Now, we divide both sides by 23 to solve for x: 299 ÷ 23 = xFind Total Expenses and Receipts: Calculating the division gives us: x ≈ 13 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: y = 36x y = 36(13)Calculating the multiplication gives us: y = 468 So, the total expenses and receipts will both equal $468 after 13 hours.

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