The planetarium sells movie tickets to adults and students. Marsha bought 2 adult tickets and 3 student tickets for $27.50.$ Jay bought 1 adult ticket and 2 student tickets for $16.25.$

A. Write a system of equations to represent the situation.

Let $a$ and $s$ be variables representing the cost for each of the adult tickets and student tickets, respectively. Then the equation representing the total cost for Marsha's tickets is $2 a + 3 s = 27.50.$ 2a is the total cost for adult, and 3s is the total cost for student tickets. Now we will deal with Jay's tickets. The equation for Jay's total cost would be $a+2s=16.25.$ For the same reasons, a and 2s are **total** costs for Jay's adult and student tickets. The system of two fully simplified equations and two variables required to solve this question are:

$$2 a + 3 s = 27.50.$$

and

$$a+2s=16.25.$$

B. Determine the cost of each adult ticket and each student ticket. (Show your work or explain how you used a graph to determine your answer.)

We know how to solve a system of equations, either by substitution, or elimination. Elimination seems to be the most efficient, as $2 \cdot a = 2a,$ and to our convenience, $2\cdot 2s-3s=s,$ which allows us to solve our equation by multiplying $a+2s=16.25$ by two and subtracting $2 a + 3 s = 27.50.$ That yields $s=5.00,$ and thus $a=6.25.$

C. Marsha bought additional tickets for her friends that decided to come to the movie. She bought a total of 4 adult tickets and 7 student tickets. What was her total cost? (Show your work.)

From #B, we know adult tickets ($a$) cost 6.25 and student tickets ($s$) cost 5.00, so we can multiply 6.25 per 4 tickets for 25 and 7 times 5 for 35. So her total cost is** 25+35=$60 **