1. Solve Systems of Equations
The Break-Even Point
Solve Systems of Equations

2. A system of equations is a set of two or more equations that have variables in common.
The common variables relate to similar quantities. You can think of an equation as a condition imposed on one or more variables, and a system as several conditions imposed simultaneously. Remember, when solving systems of equations, you are looking for a solution that makes each equation true.
Students will have learned prior to this lesson how to solve systems of linear equations.

3. Break-even point
In business, the point at which income equals expenses is called the break-even point. When starting a business, people want to know the point a which their income equals their expenses, that’s the point where they start to make a profit. In the example in the next slide, the values of y on the blue line represent dollars made and the value of y on the dotted red line represent dollars spent.

4. Break-even point
Profit
Loss

5. Words to know… Revenue: the amount of money a business makes
Expenses: the costs incurred to make the revenue. There are two types of expenses:
fixed costs: do not change regardless of how many products are sold or services performed. Fixed costs are usually the same amount each month.
Variable costs: change depending on how many products are made or how many services are performed.
Profit: the amount of revenue you get to keep after all expenses are paid
Break-even point, or BEP, is where revenue equals expenses. At this point, there is no profit.

6. Example 1
Your family starts a bed-and-breakfast in your home. You spend $500 fixing up a bedroom to rent. Your cost for food and utilities is $10 per night. Your family charges $60 per night to rent the bedroom. Step 1: Write an equation that represents your costs.
Cost, $C (in dollars)
$10 per night
Number of nights, x
$500
Modeling

7. Example 1 (con’t)
Step 2: Write an equation that represents your revenue (income).
Revenue, R (in dollars)
$60 per night
Number of nights, x

8. Example 1 (con’t) Step 3: Solve.
*The break-even point occurs when C=R. Set the expression for C equal to the expression for R. Solve the equation for x. The solution is your break-even point.

9. Example 1 (con’t) We can also solve by using a graph.
Graph the cost equation.
Graph the revenue equation on the same plane.
Find the point of intersection of the two graphs. The x-value is the break-even point.

10. Example 1 (con’t) We can also solve by using a graph.
Graph the cost equation.
Graph the revenue equation on the same plane.
Find the point of intersection of the two graphs. The x-value is the break-even point.

11. Example 2
As a young entrepreneur, your sister decides to open her own “lemonade shop” in your front yard. You help her with the math and estimates that the lemons, sugar, and bottled water cost 5 cents per cup. Futhermore, mommy dearest is charging her a $10 rental fee for use of her front yard. Your sister sells each cup for $ What is her break-even point?
Guided practice: Give students 5-10 minutes to work with their elbow partner to solve the problem.

12. Your sister will need to sell 20 cups of lemonade to break-even.
Example 2 - Solution
C = .05x + 10
R = 0.55x
Your sister will need to sell 20 cups of lemonade to break-even.
0.05x + 10 = 0.55x
10 = 0.50x
20 = x