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

Answer 1

An equation has infinitely many solutions if every value within the solution set satisfies the equation. This typically occurs when the equation is an identity or if it represents a relationship where one side is a multiple or a factor of the other side. In other words, if the equation simplifies to a true statement regardless of the value chosen for the variables, it has infinitely many solutions.

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Answer 2

A few thoughts...

Here are some potential outcomes:

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#).

Recall that Diophantine equations—equations in which the variable values are restricted to integers or positive integers—may present a very challenging set of numbers to solve.

Euler, for instance, hypothesized that the equation:

#x^4+y^4+z^4 = w^4#

contained no non-trivial solutions; however, Noam Elkies discovered one in 1988. As a result, since any solution can be multiplied by a fourth power, there are a limitless number of non-trivial solutions.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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