Question

A group of 19 students want to see the show at the planetarium. Tickets cost $11 for each student who is a member of the planetaruim's frequent visitor program and $13 for each student who is not a member. The total cost of the student's tickets is $209.
How many students in the group are members of the frequent visitor program? How many are not members? Explain how you know.

Answer 1

The answer is all of the students are members of the frequent visitor program.

Let x be the number of students that are members of the frequent visitor program.
Let y be the number of students that are not members of the frequent visitor program.
There are in total 19 students: x + y = 19

x students will pay $11, y students will pay $13 for the ticket. Totally, it will be $209: 11x + 13y = 209

Now, we have the system of two equations:
x + y = 19
11x + 13y = 209
____________
Use the method of substitution:
x + y = 19 
⇒ x = 19 - y
____________
11(19-y) + 13y = 209
11·19 - 11·y + 13y = 209
209 - 11y + 13y = 209
____________
Rearange:
13y - 11y = 209 - 209
2y = 0
⇒ y = 0

Since y = 0, x = 19:
x + y = 19
x + 0 = 19
x = 19
This means that all of the students in the group are members of the frequent visitor program.