Reduction of Equations by Multiplication.

176. The unknown quantity, instead of being connected with a known quantity by the sign + or -, may be divided by it, as in the equation x/a = b.

Here the reduction cannot be made, as in the preceding instances, by transposition. But if both members be multiplied by a, (Art. 167,) the equation will become
         x = ab.

For a fraction is mmUtplied into its denominator, by removing the denominator. This has been proved from the properties of fractions. (Art. 156.) It is also evident from the sixth axiom.

Thus x = ax/a = 3x/3 = [(a + b)x]/(a + b) = (dx + 5x)/(d + 5). For in each of these instances, x is both multiplied and divided by the same quantity; and this makes no alteration in the value. Hence,

177. When the unknown quantity is divided by a known quantity, the equation is reduced by multiplying each side by this known quantity.

The same transpositions are to be made in this case, as in the preceding examples. It must be observed also, that every term of the equation is to be multiplied. For the several terms in each member constitute a compound multiplicand, which is to be multiplied according to Art. 96.

Ex. 1. Reduce the equation      x/c + a = b + d
Multiplying both sides by      c
The product is        x + ac = bc + cd
And         x = bc + cd - ac.

2. Reduce the equation      x/(a + b) + d = h
Multiplying by a + b      x + ad + bd = ah + bh.
And         x = ag + bh - ad - bd.

178. When the unknown quantity is in the denominator of a fraction, the reduction is made in a similar manner, by multiplying the equation by this denominator.

Ex. 3. Reduce the equation      6/(10 - x) + 7 = 8
Multiplying by 10 - x       6 + 70 - 7x = 80 - 8x
And          x = 4.

179. Though it is not generally necessary, yet it is often convenient, to remove the denominator from a fraction consisting of known quantities only. This may be done, in the same manner, as the denominator is removed from a fraction, which contains the unknown quantity.

Take for example      x/a = d/b + h/c
Multiplying by a      x = ad/b + ah/c
Multiplying by b      bx = ad + abh/c
Multiplying by c      bcx = acd + abh.

Or we may multiply by the product of all the denominators at once.

In the same equation      x/a = d/b + h/c
Multiplying by abc      abcx/a = abcd/b + abch/c

Then by cancelling from each term, the letter which is common to its numerator and denominator, (Art. 142,) we have      bcx = acd + abh, as before. Hence,

180. An equation may be cleared of fractions by multiplying each side into all the denominators.

In clearing an equation of fractions, it will be necessary to observe, that the sign - prefixed to any ft action, denotes that the whole value is to be subtracted, (Art. 139,) which is done by changing the signs of all the terms in the numerator.

The equation      (a - d)/x = c - (3b - 2hm - 6n)/r
is the same as      ar - dr = crx -3bx + 2hmx + 6nx.

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