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4 x; and so the sign of each term of the numerator of the fraction

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must be changed when the denominator 2 is removed; for

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102. To reduce an equation of the first degree containing but one unknown quantity, we deduce from the preceding examples the following

RULE.

Clear the equation of fractions, if necessary.

Transpose the known terms to one member and the unknown to the other, and reduce each member to its simplest form.

Divide both members by the coefficient of the unknown quantity.

NOTE 1. To verify an equation, we have only to substitute in the equation the value of the unknown quantity found by reducing the equation. For instance, in Ex. 2, Art. 101, by substituting 20

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= 26.

When answers are not given, the work should be veri

103. Since the relations between quantities in Algebra are often expressed in the form of a proportion, we introduce here the necessary definitions.

104. RATIO is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind.

Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written, a : b, ; read, a is to b, or a divided by b.

or

a

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105. PROPORTION is an equality of ratios.

Four quan

tities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (=) or by four dots (::).

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The first and fourth terms of a proportion are called the extremes, and the second and third the means.

106. In a proportion the product of the means is equal to the product of the extremes.

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A proportion is an equation; and making the product of the means equal to the product of the extremes is merely clearing the equation of fractions.

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Multiplying by bch, chx+abch=bcx-bh x + b c d h Transposing, chx-bex+bh x-bcdh-abch Factoring 1st mem., (ch-bc+bh) x = b c d h-abch

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x

11 x

24. Reduce 6x+7 — 2 = 94 — 227 +2

91

NOTE. - Before clearing of fractions, transpose 71 and unite it

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X 9

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26. Reduce ———— 21 — +3. Ans. x = 39.

2

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Ans. x7.

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