A child's movie ticket costs $3 and an adult's ticket cost $5. If $236 was collected for one show, how many of each type of ticket were sold?

Let's assign #c# to represent the number of child movie tickets sold.

Let's assign #a# to represent the number of adult movie tickets sold.

Now we can write two equations.

#3c+5a=236#

Usually, in problems like these, they will tell you the total number of people (children and adults) that went to the show so you can write down an equation that looks like:

#a+c=TOTAL#

Lets give the variable, #y#, to all the adult tickets and the variable, #x# to all the child tickets.

Total sales: #3xxx+5xxy=236#

We are not told the total number of tickets sold, which would have been useful. However, we know that only positive integers can be plugged in.

Solving the total sales equation above for #y# in terms of #x# gives,

#y=1/5(236-3xxx)#

Graphing this equation (or plotting points) allows us to find the values that are positive integer combinations. It turns out that there are #16# different possible, positive integer combinations:

#{:("Child","Adult", "Total"),(x,y, x+y),(2,46,48),(7,43, 50),(12,40, 52),(17,37, 54),(22,34, 56),(27,31, 58),(32,28, 60),(37,25, 62),(42,22, 64),(47,19, 66),(52,16, 68),(57,13, 70),(62,10, 72),(67,7, 74),(72,4, 76),(77,1, 78):}#

All of these integer combinations lie on the linear equation

graph{1/5(236-3x)[-30,90,-10,60]}

Unfortunately, without additional information, this is the best we can do!