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Clearing denominators
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Clearing denominators
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.
Consider the equation
The smallest common multiple of the two denominators 6 and 15z is 30z, so one multiplies both sides by 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.
Without loss of generality, we may assume that the right-hand side of the equation is 0, since an equation E1 = E2 may equivalently be rewritten in the form E1 − E2 = 0.
So let the equation have the form
The first step is to determine a common denominator D of these fractions – preferably the least common denominator, which is the least common multiple of the Qi.
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Clearing denominators
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.
Consider the equation
The smallest common multiple of the two denominators 6 and 15z is 30z, so one multiplies both sides by 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.
Without loss of generality, we may assume that the right-hand side of the equation is 0, since an equation E1 = E2 may equivalently be rewritten in the form E1 − E2 = 0.
So let the equation have the form
The first step is to determine a common denominator D of these fractions – preferably the least common denominator, which is the least common multiple of the Qi.