Question

How can you tell if an equation has infinitely many solutions?

Answer 1

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.