Extraneous Solutions: Rational
Extraneous solutions can arise naturally in problems involving fractions with variables in the denominator. For example, consider this equation:
To begin solving, we multiply each side of the equation by the least common denominator of all the fractions contained in the equation. In this case, the LCD is . After performing these operations, the fractions are eliminated, and the equation becomes:
Solving this yields the single solution x = −2. However, when we substitute the solution back into the original equation, we obtain:
The equation then becomes:
This equation is not valid, since one cannot divide by zero.
Because of this, the only 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. After discarding solutions that yield an invalid equation, we will have the correct set of solutions. Note that in some cases all solutions may be discarded, in which case the original equation has no solution.
Read more about this topic: Extraneous And Missing Solutions
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