Domain of a multi variable function

Question

The function is f((x,y) = 1/((25-125^(5x-9y-55)) with the answer needing to be in the form ax+by+c. The only hint I have is as following: Convert to an exponential equation of the form a^(b)=a^(c), which implies b=c.

Answer 1

125^(5x-9y-55)=(5^3(5x-9y-55))=(5²)(5^(15x-27y-167)).

This is because (5²)(5^(15x-27y-167))=(5^(15x-27y-167+2))=5^(15x-27y-165).

So we can factorise:

f(x,y)=1/(5²(1-5^(15x-27y-167))).

5^(15x-27y-167)=1 when 15x-27y-167=0. This would give us a denominator of 5²(1-1)=0 so 15x-27y-167≠0 is the domain of f(x,y). The function can be evaluated (exists) for all x,y apart from 15x-27y-167=0.