You just need one more equation, namely

D + P = X

Now substitute for D and P in that equation,
using these:

D = 0.20X
P = 8 + (0.20X)

Then you have

  D   +       P       = X
  |           |
  v           v
0.20X + [8 + (0.20X)] = X

X = 13 students.

That's not a whole number, but it IS the answer:

In the cafeteria 20% of the 13 children, or 2 children
had hot dogs and the other 10 children had pizza.

This is the correct answer, since the last part

"8 more children had pizzas than had hot dogs"

is true because 10 is 8 more than 2.

So the problem is botched because it is against
the law to cut children into fractions, and besides
when you cut them up, they can't eat pizza or hot dogs.
J
Now tell your teacher to change the 8 to a 12, making it
this instead:

In a cafeteria if 20% of the children had hot dogs and 
the others had pizza, also 12 more children had pizzas 
than hot dogs, how many children were having lunch? 

Then it will work out to exactly 20 children.  But the
8 has to be wrong.

Edwin