The admission fee at a small fair is $1.00 for children and $4.00 for adults. on a certain day, 1800 people enter the fair and $4800 is collected. how many children and how many adults attended ?

Lets x = # of children attended the fair
y = # of adults attended the fair 

x + y = 1800 so x = 1800 - y
x + 4y = 4800 so x = 4800 - 4y

So now you have
1800 - y = 4800 - 4y
4y - y = 4800 - 1800
 3y = 3000
y = 1000
x = 1800 - 1000 = 800

children: 800 and adults: 1000

double check:
800 + 1000 = 1800 (1800 people enter the fair)

x + 4y = 4800
800 + 4(1000) = 4800
800 + 4000 + 4800
4800 = 4800 ($4800 is collected)