Applications of Linear Systems: Break-Even Point
Standard
A.CED.3 Represent constraints by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Break-Even Point
Definition: When total cost (expenses) equals total sales (revenue).
Why is this important?
In business, sometimes your product will come at the expense of the company but eventually over time the product will start bringing in profit. The cost of the product and the revenue can be modeled with linear equations and where they intersect is when the company breaks even.
Example 1
pg 256 in textbook (but with different numbers)
A puzzle expert wrote a new sudoku puzzle book. His initial costs are $632. Binding and packaging each book costs $0.74. The price of the book is $4. How many copies must be sold to break even?
Step 1
Identify cost and revenue.
What's your cost?
When is the puzzle expert using his own money? He spent $632 initially and spends $0.74 to make each book.
What's your revenue?
When is the puzzle expert earning money? He gets money for each book sold at $4.
Step 2
Come up with equations for these two scenarios!
Cost equation
When you see the word "each", you're going to multiply something by x. We said that the puzzle expert spends $0.74 on each book. In math notation, this will look like 0.74x, where x represents some number of books. Now we have that initial cost of $632 which we will simply add to the equation. It's just a one time cost that happened.
Our equation:
y = 0.74x + 632
Revenue equation
This has the word "each" again. So we're going to have 4x in our equation since it's $4 for each book. There's nothing else to add onto it so that's just it!
Our equation:
y = 4x
Step 3
Find the solution
We have two equations that make up a system of equations. What have we been learning so far? We've learned to solve these systems.
We can solve this using graphing, substitution, or elimination but let's go with graphing so that you can see visually why the break-even point is important.
The red graph, your cost equation, is y = 0.74x + 632
The blue graph, your revenue equation, is y = 4x
Hover over and click the intersection point. What's your point there? And what does it mean?
Step 4
Interpret your findings
What did the x variable represent?
- The x variable is the number of books.
What did the y variable represent?
- The y variable represents your cost/revenue in dollars.
If you looked at the graph and hovered over the intersection, that (x, y) pair is (193.865, 775.46). We just discussed what x and y are in this context so we can say:
At about 193.865 books, our total cost and our total revenue will be $775.46. This is our break-even point.
Some things to notice
- Notice how the revenue graph is higher than the cost graph after the break-even point.
- Notice how much steeper the revenue graph is and how much distance it puts between itself and the cost graph after the break-even point.
- Notice how looking at it visually easily shows you when the books will start making a profit! And with the revenue graph rising faster than the cost graph, you can see how the profit will become better the more books are being sold.