dently the same thing, suppress the denominator ben in each of the fractions, the result is aenx-bcen=bdnx-bem, an equation clear of fractions. Multiplying both members of the equation by 60, the result is 40x−45=660+12x, an equation clear of fractions. If the denominators have common factors, we can simplify the above operation by reducing them to their least common denominator, which is done (see Art. 44) by finding the least common multiple of the denominators. Thus, in the equation The least common multiple of the numbers 12, 3, 8, 6 is 24, which is, therefore, the least common denominator of the above fractions, and the equation will become 10.r 32r 312 21 52x 24 24 24 24 24' Multiplying both members of the equation by 24, the result is Hence it appears that, In order to clear an equation of fractions, reduce the fractions to a common denominator, and then multiply each term by this common denominator. In the fractional terms the common denominator will be simply suppressed. 143. From what has been said above, we deduce the following general RULE FOR THE SOLUTION OF A SIMPLE EQUATION CONTAINING ONE UNKNOWN QUANTITY. 1o. Clear the equation of fractions, and perform in both members all the algebraic operations indicated. 2o. Transpose all the terms containing the unknown quantity to one member of the equation, and all the terms containing known quantities only to the other member, and reduce each member to its most simple form. 3o. We thus obtain an equation, one member of which contains the unknown quantity alone, affected with a coefficient, and the other member contains known quantities only; the value of the unknown quantity will be found by dividing the member composed of the known quantities by the coefficient of the unknown quantity. The terms containing the unknown quantity are usually collected in the first member of the equation, though they may often be more conveniently collected in the second; the second being afterward written as the first member, and the first as the second. Sometimes an equation presents itself as one of a degree higher than the first, but both members are divisible by such a power of the unknown quantity as to reduce the equation to one of the first degree. In other cases, clearing an equation of fractions reduces it, by the canceling of those terms which contain the higher powers of the unknown quantity, to the first degree. A proportion containing an unknown quantity in any of its terms can be thrown into the form of an equation by multiplying the extremes, and also the means, and setting the two products thus formed equal to each other. Verification.-Substitute 2 for x in the given equation, it becomes 19x2+13=59-4 × 2, or 38+13=59-8, an identity. Let this process be repeated in some of the following examples. Multiplying both members by 60, 30x-75-140x-200 =960-144x+120. Transposing, 30x-140x+144x=960+ 75 +200+120. 144r-120 =16 60 Reducing to least common denominator, 10, and neglecting it, we have 12-4x-4x-10=30+ 35x+300-500. adhx+bchx+bdeh=bdfhx+bdgx+bdhm. Transposing, adhx+bchx-bdfhx-bdgx-bdhm-bdeh. Transposing and simplifying, (c-ad)x=ac—3a2bc. 144. In addition to the axioms in (Art. 141) we may subjoin the following: If two equal quantities be raised to the same power, the results will be equal. If the same root of two equal quantities be extracted, the results will be equal. Hence any equation may be cleared of a single radical quantity by transposing all the other terms to the opposite side, and then raising each member to the power denoted by the index of the radical. If there be more than one radical, the operation must be repeated. Thus : |
