1. A movie theater charges \( \$ 9 \) for adults and \( \$ 7 \) for seniors. On a particular day when 331 people paid an admission, the total receipts were \( \$ 2517 \). Find how many of each type of ticket was sold. a. Clearly state what you are trying to find. b. Set up a system of equations that could be used to solve the problem. c. Solve your system algebraically. (That is, with the substitution or elimination method.) d. Solve your system graphically. Be sure to include a sketch of your system clearly showing the intersection point.

To determine how many adult and senior tickets were sold, we can define our variables as follows: Let \( x \) represent the number of adult tickets sold, and \( y \) represent the number of senior tickets sold. We are interested in finding the values of \( x \) and \( y \) such that the number of tickets sold totals to 331 and the total revenue amounts to $2517. Now, we'll set up our system of equations based on the provided information: 1. The total number of people: \[ x + y = 331 \] 2. The total amount of money earned: \[ 9x + 7y = 2517 \] To solve the system algebraically, we can utilize the substitution method. From the first equation, we can express \( y \) in terms of \( x \): \[ y = 331 - x \] Now, we substitute this expression for \( y \) into the second equation: \[ 9x + 7(331 - x) = 2517 \] \[ 9x + 2317 - 7x = 2517 \] \[ 2x + 2317 = 2517 \] \[ 2x = 200 \] \[ x = 100 \] Now that we have \( x \), we can substitute back to find \( y \): \[ y = 331 - 100 = 231 \] Thus, 100 adult tickets and 231 senior tickets were sold. For the graphical solution, you would plot the two equations: 1. \( y = 331 - x \) (This is a line with intercepts at \( 331 \) on the y-axis and \( 331 \) on the x-axis). 2. \( y = \frac{2517 - 9x}{7} \) (This line can be rewritten to find its intercepts for plotting). The intersection point of the two lines on the graph will provide the values \( x = 100 \) and \( y = 231 \), which can be visualized as the point where both ticket-selling conditions meet. Now, let's dive into some additional fun facts about the scenario! Movie theaters have been around since the 1890s when the first public films were shown. Originally called "nickelodeons," they transformed the entertainment world, leading to the elaborate cinema palaces we see today. Who would have guessed that pricing and ticket sales evolve from those humble beginnings to complex systems we explore with algebra? In today's world, savvy theaters sometimes mix things up with dynamic pricing; this means ticket prices can vary based on the day's demand, movie newness, and even seat selection! It's all in the name of revenue optimization, keeping audiences engaged while keeping theaters profitable. So, the next time you buy a ticket, remember, there’s more math behind it than you might think!