A concert venue sells student and adult tickets, modeled by the system of equations $5 x+10 y=1000$ and $x+y=200$. What does the equation $5 x+10 y=1000$ represent?
a. The total cost of hosting the concert
b. The total number of attendees at the concert
c. The total number of tickets sold
d. The total revenue from student and adult ticket sales
Answer (1)
The equation 5 x + 10 y = 1000 models the concert ticket sales.
Assume x is the number of student tickets and y is the number of adult tickets.
5 x represents the revenue from student tickets, and 10 y represents the revenue from adult tickets.
Therefore, 5 x + 10 y = 1000 represents the total revenue, and the answer is d .
Explanation
Analyze the Equations Let's analyze the given system of equations:
5 x + 10 y = 1000 x + y = 200
We need to determine what the first equation, 5 x + 10 y = 1000 , represents in the context of concert ticket sales. The second equation, x + y = 200 , likely represents the total number of tickets sold, where x is the number of student tickets and y is the number of adult tickets.
Consider the Units Let's consider the units of each term in the equation 5 x + 10 y = 1000 . If x represents the number of student tickets, then the term 5 x must represent the revenue from student ticket sales, assuming each student ticket costs 5. S imi l a r l y , i f y$ is the number of adult tickets, then the term 10 y must represent the revenue from adult ticket sales, assuming each adult ticket costs $10.
Determine the Meaning of the Equation Therefore, the left side of the equation, 5 x + 10 y , represents the total revenue from student and adult ticket sales. The right side of the equation, 1000 , represents the total revenue in dollars. Thus, the equation 5 x + 10 y = 1000 represents the total revenue from student and adult ticket sales.
Final Answer The equation 5 x + 10 y = 1000 represents the total revenue from student and adult ticket sales. Therefore, the correct answer is d.
Examples
Understanding systems of equations is very useful in real life. For example, imagine you are planning a bake sale to raise money for your school club. You need to figure out how many cookies and brownies to bake to reach your fundraising goal. If you know how much each cookie and brownie costs to make, and how much you plan to sell them for, you can set up a system of equations to determine the optimal number of each item to bake to maximize your profit. This is similar to how businesses make decisions about pricing and production to maximize their revenue.
