x dogs and y kids.


Counting the legs, we can write an equation 

    4x + 2y = 30.


Thus we have one equation in non-negative integer numbers in two unknowns.


        +------------------------------------+
        |   It has more than one solution.   |
        +------------------------------------+


The possible solutions are listed in the Table below (every dog contributes 4 legs,
every kid contributes 2 legs)


    x dogs   y kids     total legs

       7        1     7*4 + 1*2  = 30  

       6        3     6*4 + 3*2  = 30      

       5        5     5*4 + 5*2  = 30    

       4        7     4*4 + 7*2  = 30    

       3        9     3*4 + 9*2  = 30    

       2       11     2*4 + 11*2 = 30    

       1       13     1*4 + 13*2 = 30    

There are 30 legs in my backyard but I'm counting dogs and kids.How many kids and dogs are in my backyard?

Let number of dogs, and kids, be D and K, respectively
As dogs have 4 legs, kids 2, and with a total 30 legs in the backyard, the following equation is derived: 
             4D + 2K = 30
           2(2D + K) = 2(15)
              2D + K = 15
                  2D = 15 - K
Number of dogs, or D = 
As such, 15 - K is a non-NEGATIVE-INTEGER-MULTIPLE of 2, and so, K can ONLY be ODD, natural numbers from
1 to 13 (1,3,5,7,9,11, and 13). Therefore, unless some other info or clue is provided, there could be 7 
DIFFERENT combinations of dogs and kids in the backyard. 

A clue, such as, a COMPOSITE number of kids were present, would result in the number of kids being 9 (9 being the
ONLY composite number in the list of possibilities). It follows then, that there’d be 3 dogs, which makes sense,
since it’s more logical and “normal” to have 9 kids around 3 dogs (9:3, or 3:1 kids-to-dog ratio), as opposed to
1 kid around 7 dags (1:7 kid-to-dogs ratio), or 3 kids around 6 dogs (3:6, or 1:2 kid-to-dogs ratio). 

Just SAYING!!