Suppose you are the student council member who is responsible for planning
a student dinner dance. Plans include hiring a band and buying and serving
dinner.You want to keep the ticket price as low as possible to encourage
student attendance.
As you work through the following activities, you will use systems of
equations to analyze costs and make decisions.You will write a report
detailing your choice of band, the cost of a catering service, and your ticket
price recommendation.
Task 1: Graphing
Things to consider: Consider what scales to use for the graph and look for the point of intersection.
Band A charges $600 to play for the evening. Band B charges $350 plus
$1.25 for each ticket sold.
• Write a linear equation for the cost of each band.
• Graph each equation and find the number of tickets for which the
cost of the two bands would be equal. Use this to help, Free Online Graphing Calculator
Task 2: Calculating
Things to consider: Write two equations and use elimination to solve the system.
A caterer charges a fixed cost for preparing a dinner plus an additional cost
for each person served.You know that the cost for 100 students will be $750
and the cost for 150 students will be $1050. Find the caterer’s fixed cost and
the cost per student served.
Task 3: Writing
Things to consider: Consider all expenses
Use your information from Activities 1 and 2. Assume that 200 students will
attend the dance.
• Write a report listing which band you would choose and the cost per
ticket that you need to charge to cover expenses.
• Repeat the process assuming that 300 students will attend.
Task 4: Graphing
Things to consider: Check the accuracy of your graph by calculating actual ticket prices
In Activity 3, you found two ticket prices. Each price covers the cost of the
dinner dance under certain conditions. Plan for between 200 and 300 people,
that is, x > 200 and x < 300.
• If your objective is to keep the ticket price as low as possible, even at
the risk of not covering your costs, which ticket price would you select?
Based on this choice, write a linear equation that gives the total amount
collected for ticket sales. Change your equation to an inequality to
indicate that this represents the least amount of money you expect to
collect from ticket sales.
• Now write a linear equation giving the total amount of money collected
minus all of your expenses. Change your equation to an inequality to
indicate that this represents the greatest amount of profit you expect to
collect from ticket sales.
• The two inequalities you have written, along with x > 200 and x < 300,
form a system of linear inequalities. Graph this system to show the total
amount received from ticket sales.
Finishing the Project
Things to consider: Does your analysis show clear and convincing evidence that hiring one of the two bands would be more cost-effective than hiring the other? How might your conclusions change if it is determined that only 100 students will attend the dinner dance? What if 500 students attend?
The answers to the four activities should help you complete your project.
Your report should include your analysis of the cost for dinner and
each band, depending on how many people buy tickets. Include your
recommended ticket price and note any conditions under which this ticket
price leads to a loss for the event. Illustrate your reasoning with graphs of
linear equations and inequalities.
Reflect and Revise
Present your analysis of this data to a small group of classmates. After you
have heard their analyses and presented your own, check to see that your
work is complete, clear, and convincing. If necessary, make changes to
improve your presentation.
Extending the Project
Consider other expenses you could expect to have in planning and holding
this dinner dance. Estimate the additional expenses and change your
recommended ticket price as necessary.