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If the unknown quantity, in any equation, be multiplied by any number, or quantity, the multiplier may be taken away, by dividing all the rest of the terms by it; and if it be divided by any number, the divisor may be taken away, by multiplying all the other terms by it.
Any equation may be cleared of fractions, by multiplying each of its terms, successively, by the denominators of those fractions, or by multiplying both sides by the product of all the denominators, or by any quantity that is a multiple of them.
And, if #4 – 10; then, multiplying by 12, (which z
is a multiple of 4 and 6,) 32-1-2x=120, or 5x=120, or 120 ac- - -24. 5
. It also appears, from this rule, that if the same number, or quantity, be found in each of the terms of an equation, either as a multiplier or divisor, it may be expunged from all of them, without altering the result.
Thus, if az-ab-Hac ; then, by cancelling, w=b+c.
If the unknown quantity, in any equation, be in the form of a surd, transpose the terms so that this may stand alone, on one side of the equation, and the remaining terms on the other (by Case 1); then involve each of the sides to such a power as * with the index of
Any analogy, or proportion, may be converted into an equation, by making the product of the two extreme terms equal to that of the two means. o
Thus, if 32 : 16 :: 5 : 6; then 32 ×6=16×5, or