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ca-C) ca+c) a”_ 16. Multiply together
and a’+2ac+ch a'—2acte
ax (a+bc)(a–b+c) c+ba 17. Multiply abc (0—6-a)(6_ama)
133. Any fraction may be divided by an entire quantity in two ways :
1st. By dividing its numerator; or
We may, however, derive a general rule for the division of fractions, from the following example :
1. Divide if by å
By inspecting this result, we find that the new numerator may be obtained, by multiplying the numerator of the dividend by the denominator of the divisor; and the new denominator may be obtained, by multiplying the denominator of the dividend by the numerator of the divisor. Hence the
RULE. I. Reduce entire and mixed quantities to fractional forms.
II. Invert the terms of the divisor, and proceed as in multiplication.
REDUCTION OF COMPLEX FORMS.
134. A fraction is said to be simple, when both numerator and denominator are entire; otherwise it is said to be complex.
135. To reduce a complex to a simple fraction, we may regard the quantity above the line as a dividend, and the quantity below it as a divisor, and proceed according to the last rule.
A more convenient method may be derived from the following observations :
1.-If a fraction be multiplied by its own denominator, the product will be the numerator.
2.—If a fraction be multiplied by any multiple of its denominator, the product must be entire.
Hence to simplify a complex fraction, we have the following
RULE. Multiply both numerator and denominator by the least common multiple of the denominators of the fractional parts.
EXAMPLES FOR PRACTICE.
1 1. Simplify
atx Multiplying both numerator and denominator by (a—x) (a+x), or by its equal a’--x®, we have
136. An Equation is an expression of equality between two quantities. Thus,
x+y=a is an equation, signifying that the sum of x and y is equal to a.
137. The First Member of an equation is the quantity on the left of the sign of equality; and
The Second Member is the quantity on the right of the sign of equality. Thus, in the equation,
2-3y=a-, the first member is x—3y, and the second member is a–b. The two members are sometimes called the two sides of the equation.
138. It is important to observe that the kind of equality subsisting in an equation is algebraic; that is, the two members must have the same essential sign, as well as the same arithmetical value.
139. The Unknown Quantity of an equation is the letter to which some particular value or values must be given, in order that the statement contained in the equation may be true. And such value or values are said to satisfy the equation. An equation may contain two or more unknown quantities.
140. A Root of an equation is any value which, being substituted for the unknown quantity, will satisfy the equation. For ex. ample, in