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

Question

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

Gabby's Bakery recently spent a total of $420 on new equipment, and their average hourly operating costs are $7. Their average hourly receipts are $13. The bakery will soon make back the amount it invested in equipment. What would the total expenses and receipts both equal? How many hours will that take?

The bakery's expenses and receipts will both total $__ after __ hours.

Answer 1

Hi John!

Hopefully I can help. Whenever I solve math problems of any level, here are the steps I like to take.

Q1) What is the question looking for?

A1) So depending on the type of problem this can be either easy or hard. If a question gives you an equation and says "solve for x" we can pretty easily point to what we need to find. However, for word problems we often have to decipher what we need to find and put it in math terms.

So it asks when the total expenses and receipts are equal. Lets call the expenses "E" and the receipts "R". This amounts to finding the number where E=R. Lets just stick with this for now

The second questions asks for how many hours this will take. Let's call the amount of time this takes "t" and we will have to solve for this too.

Q2) Now that we know what we need to find. What method do we need to take?

A2) So we have 3 unknowns here, so the rules of algebra say we need 3 equations in terms of our unknowns (or a "system of equations") to find these three unknowns. This is a powerful rule that helps me all the time and once you realize it you see it everywhere.

So lets start finding equations. How can we find the total expense? Well, we know she spent $420 on the equipment, and it will cost an additional $7*(hours of use). In terms of our variables we can write this as:

E = 420 + 7t

Lets do a similar thing for the receipts. She $13*(hours of use) from her equipment, so the total receipt is:

R = 13t

If you tried to find all three from these equations we couldn't, because we have three unknowns and two equations remember? So what is the last part we need? It is kind of tricky, but since we are looking at when the expenses equal the receipts it becomes pretty simple.

E = R

There we have it. 3 unknowns and 3 equations, so we should be able to solve for E, R, and t using algebraic manipulation. I will let you try and solve it from here so you get practice and learn yourself. If you need more help from here let me know!

- Evan

Answer 2

420 + 7h = 13h

420 = 6h

h=70

will pay it off after 70 hours