Rick took his family to a major league baseball game. he gave his son $100 to buy food and drinks for the group. hot dogs cost $3.50 each, milk shakes costs $8.00 each, and soda costs $5.00 each. his son returned with $25.50 in change. there were 8 beverages total in the order. how many hot dogs, milk shakes, and sodas were purchased

Rick gave his son $100 to buy food and drinks. His son returned with $25.50 in change. Therefore the amount spent was
$100 - $25.50 = $74.50.

Let
x = number of hot dogs bought (at $3.50 per hot dog).
y = number of milk shakes bought (at $8 per milk shake).
There were 8 orders of soda at $5 per soda.

The total order is $74.50, therefore
3.50x + 8y + 8*5 = 74.50
Simplify to obtain
8y = 74.5 - 3.5x
  y = 4.3125 - 0.4375x

We must have whole numbers (integers) for x and y.
Create a table that varies x from 0 to 8, and look for y to be an integer, as shown below.

  x       y
---  
 0   4.3125
 1    3.8750
 2   3.4375
 3   3
 4   2.5625
 5   2.1250
 6   1.6875
 7    1.2500
 8    0.8125

Because we cannot have fractional values for either x or y, the solution is x=3, y=3

3 hot dogs, 3 milk shakes, and 8 sodas.