In
mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an
equation equating two expressions that each are a sum of
rational expressions – which includes simple
fractions.
Example
Consider the equation
:
The
smallest common multiple of the two denominators 6 and 15''z'' is 30''z'', so one multiplies both sides by 30''z'':
:
The result is an equation with no fractions.
The simplified equation is not entirely equivalent to the original. For when we substitute and in the last equation, both sides simplify to 0, so we get , a mathematical truth. But the same substitution applied to the original equation results in , which is
mathematically meaningless.
Description
Without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, we may assume that the
right-hand side of the equation is 0, since an equation may equivalently be rewritten in the form .
So let the equation have the form
:
The first step is to determine a common denominator of these fractions – preferably the
least common denominator, which is the
least common multiple of the .
This means that each is a
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
of , so for some expression that is not a fraction. Then
:
provided that does not assume the value 0 – in which case also equals 0.
So we have now
:
Provided that does not assume the value 0, the latter equation is equivalent with
:
in which the denominators have vanished.
As shown by the provisos, care has to be taken not to introduce
zeros of – viewed as a function of the
unknowns of the equation – as
spurious solutions.
Example 2
Consider the equation
:
The least common denominator is .
Following the method as described above results in
:
Simplifying this further gives us the solution .
It is easily checked that none of the zeros of – namely , , and – is a solution of the final equation, so no spurious solutions were introduced.
References
* {{cite book , title=Algebra: Beginning and Intermediate , edition=3 , author=Richard N. Aufmann , author2=Joanne Lockwood , page=88 , publisher=Cengage Learning , year=2012 , isbn=978-1-133-70939-8
Elementary algebra
Equations