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(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.
