Agroup 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.

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.