Become math lovers!

A break-even point is the point at which the income of a business is equal to the expenses of the business. The graph below shows the break-even point with a **black dot** for one business.

Notice that the values of y on the green line represent dollars received as income and the values of y on the red line represent dollars spent on expenses. Therefore, y is used to represent both income and expenses.

**Example:**

To run a business, you need to spend 2 dollars for each item sold plus 100 dollars in extra fees. The selling price of each item is 3 dollars. What is the break-even point?

Let x be the number of items sold and let y be the amount of dollars of income and expenses.

Income = selling price times number of items sold

y = 3x

Expenses = cost for the number of items sold plus cost of extra fees

y = 2x + 100

The break-even point occurs when the income of the business is equal to the expenses of the business.

3x = 2x + 100

3x - 2x = 2x - 2x + 100

x = 100

The break-even point for this business occurs when you sell 100 items

y = 3x = 3(100) = 300

y = 2x + 100 = 2(100) + 100 = 200 + 100 = 300