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, xta 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 (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. 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 Take the equation 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 8x – 9x + 2x = 132. 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. Exercises. Clear the following equations of fractions : – 1.+ – 4= 3. Ans. 7x + 5x — 140 = 105. 2. 6+6 -= 8. Ans. 9x + 6x — 2x = 432. C018 = 20. Ans. 18x + 12x - 4x + 3x=720. 18 ti Ans. 14 x + 10x — 35x = 280. 1-%+= 15. Ans. 15 x — 122 + 10x = 900. 2 = 12. Ans. 18x – 12 x + 9x + 8x = 864. 9.-&+f=g. Ans. ad – be + ödf = bdg. 10. aut et + 4a = 40e* _ 20 – 38. a The L. C. M. of the denominators is a'b?. a+bx – 2a2bc2x + 4 a+b2 = 463c2x – 5 a6 + 2 a?B?c? – 3 a365. · 105. Transposition is the operation of changing a term from one member to the other without destroying the equality of the members. Take, for example, the equation 5x – 6= 8+2x. If, in the first place, we subtract 2x from both members, the equality will not be destroyed, and we have 5x – 6 – 2x = 8. Whence we see that the term 2x, which was additive in the second member, becomes subtractive by passing into the first. In the second place, if we add 6 to both members of the last equation, the equality will still exist, and we have 5x — 6 - 2x+6=8+6; or, since — 6 and + 6 cancel each other, we have 5x – 2x = 8+6. Hence the term which was subtractive in the first member, passes into the second member with the sign of addition. 106. Therefore, for the transposition of the terms, we have the following rule: Any term may be transposed from one member of an equation to the other, if the sign be changed. |