Here is a summary of the scenario your classmate presented for the talent show:
| Main show | The main show will last two hours and will include twelve individual acts and six group acts. |
| --------- | ------------------------------------------------------------------------------------------ |
| Final show | The final show will last 30 minutes and will include the top four individual acts and the top group act. |
The equations he came up with are:
[tex]
\begin{array}{l}
12 x+6 y=120 \\
4 x+y=30
\end{array}
[/tex]
1. What do [tex]$x$[/tex] and [tex]$y$[/tex] represent in this situation?
Answer (2)
x represents the time in minutes allocated to each individual act.
y represents the time in minutes allocated to each group act.
The equation 12 x + 6 y = 120 represents the main show's duration.
The equation 4 x + y = 30 represents the final show's duration.
b o x e d x = time for individual act , y = time for group act
Explanation
Problem Analysis Let's analyze the given equations in the context of the talent show scenario.
Analyzing the First Equation The first equation is 12 x + 6 y = 120 . Here, 12 represents the number of individual acts, and 6 represents the number of group acts in the main show, which lasts 120 minutes (2 hours). Thus, x likely represents the time allocated to each individual act, and y represents the time allocated to each group act, both measured in minutes.
Analyzing the Second Equation The second equation is 4 x + y = 30 . In the final show, there are 4 individual acts and 1 group act, with the show lasting 30 minutes. This reinforces the idea that x represents the time in minutes for each individual act, and y represents the time in minutes for each group act.
Conclusion Therefore, x represents the time in minutes allocated to each individual act, and y represents the time in minutes allocated to each group act.
Examples
Understanding what variables represent in a system of equations is crucial in many real-world scenarios. For instance, if you're planning a school event with different activities, you might use equations to determine how much time to allocate to each activity based on the total event time and the number of participants. Similarly, in budgeting, you can use variables to represent costs and resources, helping you make informed decisions about resource allocation. This skill is also valuable in project management, where you need to balance time, resources, and tasks to achieve project goals efficiently. By mastering these concepts, you'll be well-equipped to tackle various planning and resource management challenges.
In this talent show scenario, x represents the time in minutes for each individual act, while y is the time in minutes for each group act. The equations describe how these times contribute to the total durations of the main and final shows. Understanding these variables helps in effectively organizing the schedule for the performances.
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