mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

equating two expressions that each are a sum of rational expressions – which includes simple

fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...

Example

Consider the equation :

\frac x 6 + \frac y  = 1.

The smallest common multiple of the two denominators 6 and 15''z'' is 30''z'', so one multiplies both sides by 30''z'': :

5xz + 2y = 30z. \,

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 :

\sum_^n \frac = 0.

The first step is to determine a common denominator of these fractions – preferably the

least common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The lo ...

, which is the

least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...

of the . This means that each is a factor of , so for some expression that is not a fraction. Then :

\frac = \frac = \frac D \,,

provided that does not assume the value 0 – in which case also equals 0. So we have now :

\sum_^n \frac = \sum_^n \frac D = \frac 1 D \sum_^n R_i P_i = 0.

Provided that does not assume the value 0, the latter equation is equivalent with :

\sum_^n R_i P_i = 0\,,

in which the denominators have vanished. As shown by the provisos, care has to be taken not to introduce

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

s of – viewed as a function of the

unknown Unknown or The Unknown may refer to: Film and television Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), Australian silent film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film), American silent drama ...

s of the equation – as spurious solutions.

Example 2

Consider the equation :

\frac+\frac-\frac = 0.

The least common denominator is . Following the method as described above results in :

(x+2)+(x+1)-x = 0.

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