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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Cathryn Jackson*

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Businesses use breakeven points to determine price and sell products. Learn how to use systems of linear equations with revenue and cost functions to find the breakeven point.

Max wants to start his own lemonade stand. It costs him $3 to make each cup of lemonade, plus he has to pay a $25 renter's fee for his stand. He sells the lemonade for $5 per cup. Max wants to know how many cups of lemonade he has to sell before he can start making a profit.

Max needs to find his breakeven point. In algebra, the **breakeven point** is the point where two linear functions intersect. In marketing, this point represents the point where products neither make a profit nor incur a loss. For Max to start earning a profit, he needs to find his breakeven point by using two linear functions: **cost function** and **revenue function**.

The **cost function** is the linear function that represents the seller's cost of a product. Max spends $3 producing each cup of lemonade, and he has a renter's fee for his stand. We can use this information to create our first linear function. The cost function is *C(x) = mx + b*. You might recognize this as the slope-intercept formula in algebra.

In this function, the *C(x)* is the total cost of the product. That's why it's called the cost function. The *m* is the variable cost, which in this case is $3. The reason *m* is considered the variable cost is because it depends on how many cups of lemonade Max is making. It is also a variable because, depending on the product, the cost of the product can potentially change. For example, if Max is making 10 cups of lemonade, then it will cost him $30, not including the cost of his renter's fee.

The *b* or *y*-intercept in this function represents the fixed cost. This is the cost that does not change regardless of how many cups of lemonade Max makes. In this case, the fixed cost would be Max's renter's fee. Therefore our cost function would look like this - *C(x)* = 3*x* + 25 - and our graph would look like this.

We aren't done yet! We know the cost of the product, but we don't know how much Max can make off of the product. We can find this information using the ** revenue function**, which is the linear function that represents the seller's gross income from a product. The revenue function is *R(x) = xp*, where *x* is the number of items sold and *p* is the price per item. This income is only how much Max gets back in total from the product. In this case, Max is selling each cup of lemonade for $5. Therefore, our revenue function would look like this - *R(x)* = *x* * 5 - and our graph would now look like this.

You can find the breakeven point of two linear functions either graphically or algebraically. Take a look at our final graph.

See where the two lines intersect? This is the breakeven point and literally where Max will break even in his lemonade business. Everything to the left of this point, between the two lines, represents a loss, and everything to the right of this point, between the two lines, represents a profit. These two lines intersect at the point (12.5, 62.5). Therefore, Max must sell 12.5 cups of lemonade before he can make a profit. The 62.5 is the total cost - 12.5 cups of lemonade cost $37.50 to make plus the $25 renter's fee.

Max also wants to find his breakeven point algebraically. Remember that *x* is the number of items sold. Therefore we can set the cost function equal to the revenue function, like this - 3*x* + 25 = 5*x* - and solve for *x*.

3*x* + 25 = 5*x*

12.5 = *x*

You can put 12.5 back into one of the functions to find Max's total cost:

*R(x)* = 12.5(5)*R(x)* = 62.5

Max will have to sell 12.5 cups of lemonade to break even, and it will cost him $62.50.

Max has learned a lot about pricing a product. First, he had to find the **breakeven point** of his product, which is the point where two linear functions intersect and, in marketing, represents the point where products neither make a profit nor incur a loss. He found the breakeven point by using the **cost function** and the **revenue function** of his lemonade. The **cost function** is the linear function that represents the seller's cost of a product, while the **revenue function** is the linear function that represents the seller's gross income from a product. Now Max knows exactly how much lemonade he needs to sell to earn a profit.

When you are finished watching this lesson, you might:

- Define breakeven point and cost and revenue function
- Calculate the cost and revenue function and the breakeven point of a product

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- Writing & Evaluating Algebraic Expressions for Two-Dimensional Geometric Figures 4:10
- Writing & Evaluating Real-Life Linear Models: Process & Examples 11:00
- Applying Systems of Linear Equations to Breakeven Point: Steps & Example 5:44
- Using Quadratic Models to Find Minimum & Maximum Values: Definition, Steps & Example 9:54
- Go to Mathematical Modeling

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