Both situations frequently result from performing operations that are not invertible for some or all values of the variables involved, which prevents the chain of logical implications from being bidirectional.
One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions.
However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before.
For example, consider the following equation: If we multiply both sides by zero, we get, This is true for all values of
, so the solution set is all real numbers.
But clearly not all real numbers are solutions to the original equation.
The problem is that multiplication by zero is not invertible: if we multiply by any nonzero value, we can reverse the step by dividing by the same value, but division by zero is not defined, so multiplication by zero cannot be reversed.
More subtly, suppose we take the same equation and multiply both sides by
This counterintuitive result occurs because in the case where
multiplies both sides by zero, and so necessarily produces a true equation just as in the first example.
In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero.
But it is not sufficient to exclude these values, because they may have been legitimate solutions to the original equation.
For example, suppose we multiply both sides of our original equation
Extraneous solutions can arise naturally in problems involving fractions with variables in the denominator.
After performing these operations, the fractions are eliminated, and the equation becomes: Solving this yields the single solution
is extraneous and not valid, and the original equation has no solution.
However, it is not always simple to evaluate whether each operation already performed was allowed by the final answer.
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation.
However, more insidious are missing solutions, which can occur when performing operations on expressions that are invalid for certain values of those expressions.
For example, if we were solving the following equation, the correct solution is obtained by subtracting
: By analogy, we might suppose we can solve the following equation by subtracting
, which involves the indeterminate operation of dividing by zero when
It is generally possible (and advisable) to avoid dividing by any expression that can be zero; however, where this is necessary, it is sufficient to ensure that any values of the variables that make it zero also fail to satisfy the original equation.
For example, suppose we have this equation: It is valid to divide both sides by
, obtaining the following equation: This is valid because the only value of
is zero or negative, because this can only remove solutions we do not care about.
Multiplication and division are not the only operations that can modify the solution set.
is actually not in general the positive square root of
is negative, the positive square root of
We can also modify the solution set by squaring both sides, because this will make any negative values in the ranges of the equation positive, causing extraneous solutions.