is equivalent to multiplying by its reciprocal. But the re ciprocal of a fraction, a d is (§ 28): consequently, to divide с any quantity by a fraction, we invert the terms of the divisor, and multiply by the resulting fraction. Hence Whence the following rule for dividing one fraction by another : Reduce mixed quantities to fractional forms. Invert the terms of the divisor, and multiply the dividend by the resulting fraction. NOTE. The same remarks as were made on factoring and reducing, under the head of "Multiplication," are applicable in division. CHAPTER V. EQUATIONS OF THE FIRST DEGREE, AND INEQUALITIES. EQUATIONS OF THE FIRST DEGREE. 95. An equation is an expression of equality between two quantities. Thus, x=b+c is an equation expressing the fact that the quantity x is equal to the sum of the quantities b and c. 96. Every equation is composed of two parts, connected by the sign of equality. These parts are called members. The part on the left of the sign of equality is called the first member; that on the right, the second member. Thus, in the equation x + a=b-c, x+a is the first member; and b C, the second member. 97. An equation of the first degree is one which involves only the first power of the unknown quantity. Thus, 6x+3x-5=13 ax + bx + c = d (1) (2) are equations of the first degree. 98. A numerical equation is one in which the coefficients of the unknown quantity are denoted by numbers. 99. A literal equation is one in which the coefficients of the unknown quantity are denoted by letters. Equation (1) is a numerical equation; Equation (2) is a literal equation. 102 Transformation of Equations. 100. The transformation of an equation is the operation of changing its form without destroying the equality of its members. 101. An axiom is a self-evident proposition. 102. The transformation of equations depends upon the following axioms: Axiom 1. If equal quantities be added to both members of an equation, the equality will not be destroyed. Axiom 2. If equal quantities be subtracted from both members of an equation, the equality will not be destroyed. Axiom 3. If both members of an equation be multiplied by the same quantity, the equality will not be destroyed. Axiom 4. If both members of an equation be divided by the same quantity, the equality will not be destroyed. Axiom 5. Like powers of the two members of an equation are equal. Axiom 6. Like roots of the two members of an equation are equal. 103. Two principal transformations are employed in the solution of equations of the first degree, — clearing of fractions, and transposing. The L. C. M. of the denominators is 12. If we multiply both members of the equation by 12, each term will reduce to an entire form, giving Any equation may be reduced to entire terms in the same manner. 104. Hence, for clearing of fractions, we have the following rule: Find the L. C. M. of the denominators. Multiply both members of the equation by it, reducing the fractional to entire terms. NOTES.1. The reduction will be effected, if we divide the L. C. M. by each of the denominators, and then multiply the corresponding numerator by the quotient, dropping the denominator. 2. The transformation may be effected by multiplying each numerator into the product of all the denominators except its own, omitting denominators. 3. The transformation may also be effected by multiplying both members of the equation by any multiple of the denominators. |