How can you tell if an equation has infinitely many solutions?
A few thoughts...
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(x-1) = 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 non-trivial solutions, but Noam Elkies found one in 1988, hence there are an infinite number of non-trivial solutions, since any solution can be multipled by a fourth power.