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

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Consider the equation

${\frac {x}{6}}+{\frac {y}{15z}}=1.$

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

$5xz+2y=30z.\,$

The result is an equation with no fractions.

The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0, a mathematical truth. But the same substitution applied to the original equation results in x/6 + 0/0 = 1, which is mathematically meaningless.

Description

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Without loss of generality, we may assume that the right-hand side of the equation is 0, since an equation E_{1} = E_{2} may equivalently be rewritten in the form E_{1} − E_{2} = 0.

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

$\sum _{i=1}^{n}R_{i}P_{i}=0\,,$

in which the denominators have vanished.

As shown by the provisos, care has to be taken not to introduce zeros of D – viewed as a function of the unknowns of the equation – as spurious solutions.

Following the method as described above results in

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

Simplifying this further gives us the solution x = −3.

It is easily checked that none of the zeros of x(x + 1)(x + 2) – namely x = 0, x = −1, and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.

References

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Richard N. Aufmann; Joanne Lockwood (2012). Algebra: Beginning and Intermediate (3 ed.). Cengage Learning. p. 88. ISBN 978-1-133-70939-8.