Question

Andrea sells tickets at the Showtime Cinema. On Thursday night, Back to the Future was playing. Adults, senior citizens and children bought tickets at the following rates:

Adults $5.00
Senior Citizens $1.00
Children $.10

Andrea sold 100 tickets worth $100. How many adult, senior and children tickets were sold?

Answer 1

number of adults --- x
number of seniors --- y
number of children 100-x-y

5x + y + .1(100-x-x-y) = 100
times 10
50x + 10y + 100 -x-y = 1000
49x + 9y = 900

We are looking for integer solutions of x and y, (can't have partial people)

the x-intercept is appr 19
the y-intercept is appr 90

49x = 900-9y
x = (900-9y)/49

so 900-9y must be a multiple of 49

starting with trials of y = 1, 2, 3, ..
I lucked out at y = 2
when y = 2
x = 18

knowing the slope of our linear equation is -49/9
increasing the y value by 49 and decreasing the x by 9 would give us another solution.
so y = 51 and x = 9 would be another solution

so case 1:
18 adults
2 seniors
80 children
check:
-- sum is 100,
-- income = 5(18) + 1(2) + .1(80) = $100

case 2:
9 adults
51 seniors
40 children

check:
--- sum is 100
-- income = 5(9) + 1(51) + .1(40) = 100

(there is a third case of
0 adults
100 students
0 children , but it said that adults, seniors and children bought tickets , so I would exclude that)