Applying Systems of Linear Equations to Breakeven Point: Steps & Example

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  • 0:01 Linear Models and Lemons
  • 0:22 Breakeven Point
  • 4:57 Lesson Summary
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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.

Linear Models and Lemons

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.

Breakeven Point

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) = 3x + 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.


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