we have 9x+3a-(Sx-4a)=x+7a, placing this over the common denominator 12, we find

If any of the quantities to be multiplied are mixed, they must, by Case III, be reduced to a fractional form; then mul

4

tiply Together all the numcrators for a numcrator, and all the denominators together for a denominator.

EXAMPLES.

x+b

xta 1. Multiply by

2 3 The product of the numerators will be

(x+a)*(x+6)=x2 +ax+bx+ab; and the product of the denominators is 2x3=6.

xtu a+b 22 +ax+bx+ab
Hence,

Х
2
2

6
2-12 x2 +12
2. Multiply by

be
btc

24-24
Ans.

bac+bc?

3-2 4+x 3 3. What is the continued product of

and = ? 72

36-3x43x2 Ans.

98

a+ba-b 4. What is the product of

? 2

a2-62 Ans.

4

2x h-d 6 5. What is the continued product of

and

7

a

m

C

y-1

6. What is the product of ty+1,

y
by

?
26

by? – by+y2 - 1 Ans.

CASE IX.

(65.) To divide one fraction by another.

RULE.

If there are any mixed quantities, reduce them to a fractional form, by Case III. Then invert the divisor, and multiply as in Case VIII.

If we invert the divisor, and then multiply, we have

Ans.

CHAPTER III.

SIMPLE EQUATIONS.

(66.) An equation is an expression of two equal quantities with the sign of equality placed between them.

The terms or quantities on the left-hand side of the sign of equality constitute the first member of the equation, those on the right constitute the second member. Thus, x+2=a

(1)
-1=)

(2)
2
3x+7=C

(3) are equations; the first is read “ x increased by 2 equals a.”

The second is read “one-half of x diminished by 1 equals b.”. The third is read “three times x increased by 7 equals c.”

(67.) Nearly all the operations of algebra are carried on by the aid of equations. The relations of a question or problem are first to be expressed by an equation, containing known "quantities as well as the unknown quantity. Afterwards we must make such transformations upon this equation as to bring the unknown quantity by itself on one side of the equation, by which means it becomes known.

(68.) An equation of the first degree, or a simple equation, is one, in which the unknown has no power above the first degree.

(69.) A quadratic equation is an equation of the second degree, that is, the unknown quantity has no power above the second degree.

(70.) An equation of the third, fourth, &c., degrees, is one which has no power of the unknown quantity above the third, fourth, &c., degrees.

And in general, an equation which involves the mth power of the unknown quantity, is called an equation of the mth degree.

(71.) The following axioms will enable us to make many transformations upon the terms of an equation without destroying their equality.

AXIOMS.

I. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed.

II. If equal quantities be subtracted from both members of an equation, the equality of the members will not be destroyed.

III. If both members of an equation be multiplied by the same number, the equality will not be destroyed.

IV. If both members of an equation be divided by the same number, the equality will not be destroyed.

CLEARING EQUATIONS OF FRACTIONS.

(72.) When some of the terms of an equation are fractional, it is necessary to so transform it as to cause the denominators to disappear, which process is called clearing of fractions.