a into an C axa ax3 + c4 into an at to x12 + + a 18 a14 Ex. 11. It is required to convert infinite series. a ax Ans. - + + +, &c. с c2 a2 + x2 Ex. 12. It is required to convert x infinite series. 1 Ans. &c. a a6 6 6 Ex. 13. It is required to convert or in 10- -1' to an infinite series. 6 6 6 6 Ans. +, &c. 10' 102 103 104 1 1 Ex. 14. It is required to convert into 4 5 an infinite series. Ans. 1 1 1 1 +&c. &c. 53 9' + CHAPTER III. ON SIMPLE EQUATIONS, INVOLVING ONLY ONE UNKNOWN QUANTITY. 184. In addition to what has been already said, (Art. 34), it may be here observed, that the expression, in algebraic symbols, of two equivalent phrases contained in the enunciation of a question, is called an equution, which, as has been remarked by GARNIER, differs from an equality, in this, that the first comprehends an unknown quantity combined with certain unknown quantities; where as the second takes place but between quantities dom that are known. Thus, the expression a= + 2 2 (Art. 102), according to the above remark, is called an equality ; because the quantities a, s, and do are supposed to be known. And the expression *+-x-d=s, (Art. 103), is called an equation, because the unknown quantity X, is combined with the given quantities d and s. Also, x-a=0 is an equation which asserts that x-a is equal to nothing, and therefore, that the positive part of the expression is equal to the negative part. 185. A simple equation is that, which contains only the first power of the unknown quantity, or the unknown quantity merely in its simplest form, after the terms of the equation have been properly arranged : bax= , 4 3 where x denotes the unknown quantity, and the other letters, or numbers, the known quantities. § I. REDUCTION OF SIMPLE EQUATIONS. 186. Any quantity may be transposed from one side of an equation to the other, by changing its sign. Because, in this transposition, the same quantity, is merely added to or subtracted from each side of the equation; and, (Art. 48, 49,) if equals be added to or subtracted from equal quantities, the sums or remainders will be equal. Thus, if x +5=12; by subtracting 5 from each side, we shall have x +5--5=12—5; but 5-550, and 12-5=7; hence x=7. Also, if cta=h-2x; by subtracting a from each side, we shall have ata--a=b-2x -a; and by adding 2x to each side, we shall have c+a+2=0–2x=a+2c ; but a-a=0, and —2x+20=0; therefore x + 2x=b-a, or 3x=b-a. Again, if ax-o=d, and c be added to each side. ax-c+c=d+c, or ax=d+c. Also, if 5x—7=2x+12; by subtracting %w from each side, we shall have 5x—7—2x=2x+12—2x, or 3.0-7=12; subtracting – 7, or, which is the same thing, adding +7 to each side of this last equation, and we shall have 3x −7+7=12+7; but 7–7=0,., 3x=19. Finally, if x-a+b=c-2x+d; then, by subtracting b from each side, we shall have x-at-b-b=(--2c+d--; and adding a+2x to each side, it becomes x-atb-bta+2x=(-2x+d-btat. 2x ; but a-a=0, b-b=0, and -2x + 2x=0; therefore, +2x=cta-b+d, or 3x=cta-b+d. Cor. 1. Hence, if the signs of the terms on each side of an equation be changed, the two sides still remain equal : because in this change every term is transposed: Thus, if -x+b-c=a—9+x; then, x-b+c=9-a-x; or, (which is the same thing) by transposing the right-hand side to the left and the reverse, we shall have 9-a-d=x-6+c. Cor. 2. Hence, when the known and unknown quantities are connected in an equation by the signs - + or --, they may be separated by transposing the known quantities to one side, and the unknown to the other. Thus, if 3x-9-=12+b--4x2 ; then, 4x2 + 3x=a+b+21. Also, if 3x2 -2+x=b-43-3x*; then, 3x4 + °; 4x3+3x2=b+-2. Hence also, if any quantity be found on both sides of an equation, it may be taken away from each ; thus, if ata=a+5, then x=5; if x-b= c+d-b, then x=c+d; because, by adding b to each side, we shall have 2–6+b=c7d-6+6 ; but b-b=0,.. x=c+d. 187. If every term on each side of an equation be multiplied by the same quantity, the results will be equal : because, in multiplying every term on each side by any quantity, the value of the whole side is multiplied by that quantity; and, (Art. 50), if equals be multiplied by the same quantity, the products will be equal. Thus, if x=5+a, then 6x=30+6a, by multiplying every term by 6. And, if = 4, then, multiplying each side by 2, we have X2=4x2, or x=8, because, (Art. 156), 5X2=x. . )=. Also, if 3=a4b, then, by multiplying every term by 4, we shall have x-12=4a-4b. , Again, if 2x+1=*; then, 4x−3+2=2x ; and 4x-2x=3—2, or 2x=1. 4 3 2 Cor. 1. Hence, an equation of which any part is fractional, may be reduced to an equation expressed in integers, by multiplying every term by the denominator of the fraction ; but if there be more fractions than one in the given equation, it may be so reduced by multiplying every term by the product of the denominators, or by the least common multiple of them; and it will be of more advantage, to multiply by the least common multiple, as then, the equation will be in its lowest terms. Let t+=11; then, if every term be mul 2 4 tiplied by 24, which is the product of all the deno; 2 4 24; and 12x+8 +-6x=264 ; or, if every term of the proposed equation be multiplied by 12, which is the least common multiple of 2, 3, 4, (Art. 147); we shall have 6x +4.0+30=132. an equation in its Towest terms. 3 |