let

x = the number of $12 pizzas

y = the number of $16 pizzas

x and y are positive integers

12x + 16y = 136 divide both sides by 4

3x + 4y = 34

4y = 34 - 3x

y = (34 - 3x)/4

x must be an even number (otherwise 34 - 3x will be odd and we need 34 - 3x to bi divisible by 4).

Also 34 - 3x >=0 follows x <= 34/3 or x <=11.

for x = 2

y = (34 - 3*2)/4

y = 7

one solution is: 2 veggie pizzas and 7 meaty pizzas can be ordered.

for x = 4

y = (34 - 3*4)/4

y = 22/4 is not integer thus x = 4 is not solution

for x = 6

y = (34 - 3*6)/4

y = 16/4

y = 4

second solution: 6 veggie pizzas and 4 meaty pizzas can be ordered.

for x = 8

y = (34 - 3*8)/4

y = 10/4 is not integer thus x = 8 is not solution

for x = 10

y = (34 - 3*10)/4

y = 4/4

y = 1

third solution: 10 veggie pizzas and 1 meaty pizzas can be ordered.

We examined all posibilities and found the follwoing 3 solutions:

first solution is: 2 veggie pizzas and 7 meaty pizzas can be ordered.

second solution: 6 veggie pizzas and 4 meaty pizzas can be ordered.

third solution: 10 veggie pizzas and 1 meaty pizzas can be ordered.