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The cost function    is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in [link] . The x -axis represents quantity in hundreds of units. The y -axis represents either cost or revenue in hundreds of dollars.

The point at which the two lines intersect is called the break-even point    . We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.

The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function    is the revenue function minus the cost function, written as P ( x ) = R ( x ) C ( x ) . Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

Finding the break-even point and the profit function using substitution

Given the cost function C ( x ) = 0.85 x + 35,000 and the revenue function R ( x ) = 1.55 x , find the break-even point and the profit function.

Write the system of equations using y to replace function notation.

y = 0.85 x + 35,000 y = 1.55 x

Substitute the expression 0.85 x + 35,000 from the first equation into the second equation and solve for x .

0.85 x + 35,000 = 1.55 x 35,000 = 0.7 x 50,000 = x

Then, we substitute x = 50,000 into either the cost function or the revenue function.

1.55 ( 50,000 ) = 77,500

The break-even point is ( 50,000 , 77,500 ) .

The profit function is found using the formula P ( x ) = R ( x ) C ( x ) .

P ( x ) = 1.55 x ( 0.85 x + 35 , 000 )          = 0.7 x 35 , 000

The profit function is P ( x ) = 0.7 x −35,000.

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Writing and solving a system of equations in two variables

The cost of a ticket to the circus is $ 25.00 for children and $ 50.00 for adults. On a certain day, attendance at the circus is 2,000 and the total gate revenue is $ 70,000. How many children and how many adults bought tickets?

Let c = the number of children and a = the number of adults in attendance.

The total number of people is 2,000. We can use this to write an equation for the number of people at the circus that day.

c + a = 2,000

The revenue from all children can be found by multiplying $ 25.00 by the number of children, 25 c . The revenue from all adults can be found by multiplying $ 50.00 by the number of adults, 50 a . The total revenue is $ 70,000. We can use this to write an equation for the revenue.

25 c + 50 a = 70,000

We now have a system of linear equations in two variables.

c + a = 2,000 25 c + 50 a = 70,000

In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either c or a . We will solve for a .

c + a = 2,000 a = 2,000 c

Substitute the expression 2,000 c in the second equation for a and solve for c .

  25 c + 50 ( 2,000 c ) = 70,000   25 c + 100,000 50 c = 70,000                          25 c = −30,000                                 c = 1,200

Substitute c = 1,200 into the first equation to solve for a .

1,200 + a = 2,000                a = 800

We find that 1,200 children and 800 adults bought tickets to the circus that day.

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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