First equation is 
m + n = 15.                    (1)

The second equation is

 +  = 36.   (2)  ( the number of diags of an n-sided polygon is  )

Simplify and rewrite it:

m + n = 15,                   (1')
n^2 - 3n + m^2 - 3m = 72.     (2')

Simplify and rewrite it one more time. In the second eqn. use m+n = 15:

m + n = 15,                   (1'')
n^2 + m^2 = 72+3*15.          (2'')
 
or

m + n = 15,                   (1''')
n^2 + m^2 = 117.              (2''')

Solve by the substitution method.

Answer.  1) n = 6, m = 9.   2) n = 9, m= 6.