Using Systems of Equations to Investigate Profits
Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation where quantity and price. The revenue function is shown in orange in Figure 10.
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 Figure 10. The -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 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 and the revenue function find the break-even point and the profit function.
- Answer
Write the system of equations using to replace function notation.
Substitute the expression from the first equation into the second equation and solve for
Then, we substitute into either the cost function or the revenue function.
The break-even point is
The profit function is found using the formula
The profit function is
Analysis
The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11.
We see from the graph in Figure 12 that the profit function has a negative value until when the graph crosses the x-axis. Then, the graph emerges into positive y-values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.
Writing and Solving a System of Equations in Two Variables
The cost of a ticket to the circus is for children and for adults. On a certain day, attendance at the circus is and the total gate revenue is How many children and how many adults bought tickets?
- Answer
Let c = the number of children and a = the number of adults in attendance.
The total number of people is We can use this to write an equation for the number of people at the circus that day.
The revenue from all children can be found by multiplying by the number of children, The revenue from all adults can be found by multiplying by the number of adults, The total revenue is We can use this to write an equation for the revenue.
We now have a system of linear equations in two variables.
In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either or We will solve for
Substitute the expression in the second equation for and solve for
Substitute into the first equation to solve for
We find that children and adults bought tickets to the circus that day.
Meal tickets at the circus cost for children and for adults. If meal tickets were bought for a total of how many children and how many adults bought meal tickets?
Access these online resources for additional instruction and practice with systems of linear equations.
9.1 Section Exercises
Verbal
1.
Can a system of linear equations have exactly two solutions? Explain why or why not.
2.
If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.
3.
If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?
4.
If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?
5.
Given a system of equations, explain at least two different methods of solving that system.
Algebraic
For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
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For the following exercises, solve each system by substitution.
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For the following exercises, solve each system by addition.
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For the following exercises, solve each system by any method.
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Graphical
For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
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Technology
For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
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Extensions
For the following exercises, solve each system in terms of and where are nonzero numbers. Note that and
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Real-World Applications
For the following exercises, solve for the desired quantity.
56.
A stuffed animal business has a total cost of production and a revenue function Find the break-even point.
57.
A fast-food restaurant has a cost of production and a revenue function When does the company start to turn a profit?
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A cell phone factory has a cost of production and a revenue function What is the break-even point?
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A musician charges where is the total number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
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A guitar factory has a cost of production If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.
For the following exercises, use a system of linear equations with two variables and two equations to solve.
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Find two numbers whose sum is 28 and difference is 13.
62.
A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.
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The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?
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A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?
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A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?
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276 students enrolled in a freshman-level chemistry class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.
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There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?
68.
A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.
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If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?
70.
An investor earned triple the profits of what she earned last year. If she made $500,000.48 total for both years, how much did she earn in profits each year?
71.
An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110% increase on the money she invested. On the second investment, Riverside Community, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?
72.
If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the other that pays interest, and the investor earns $737.50 annual interest, how much was invested in each account?
73.
If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?
74.
CDs cost $5.96 more than DVDs at All Bets Are Off Electronics. How much would 6 CDs and 2 DVDs cost if 5 CDs and 2 DVDs cost $127.73?
75.
A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $6,404.40, how many of each type of sneaker were sold?
76.
A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $5,075 was taken in. How many student tickets and adult tickets were sold?
77.
Admission into an amusement park for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket?