How can you tell if an equation has infinitely many solutions?
1 Answer
A few thoughts...
Explanation:
Here are a few possibilities:

The equation simplifies to the point that it no longer contains a variable, but expresses a true equation, e.g.
#0 = 0# . For example:#2x+2 = 2(x+1)# simplifies in this way. 
The equation has an identifiable solution and is periodic in nature. For example:
#tan^2 x + tan x  5 = 0# has infinitely many solutions since#tan x# has period#pi# . 
The equation has a piecewise behaviour and simplifies within at least one of the intervals to a true equation without variables. For example:
#abs(x+1)+abs(x1) = 2# , which simplifies suitably for#x in [1, 1]# . 
The equation has more than one variable and does not force uniqueness. For example:
#x^2+y^2=1# has infinitely many solutions, but#x^2+y^2=0# has one solution (assuming#x, y in RR# ).
Note that it may be extremely difficult to determine the number of solutions in the case of Diophantine equations  equations where the values of the variables are limited to integers or to positive integers.
For example, Euler conjectured that the equation:
#x^4+y^4+z^4 = w^4#
had no nontrivial solutions, but Noam Elkies found one in 1988, hence there are an infinite number of nontrivial solutions, since any solution can be multipled by a fourth power.