is often found, as has been observed by EULER in his Algebra, to be of the greatest utility; it is also remarkable, that an infinite series, though it never ceases, may have a determinate value. It should likewise be observed, that from this branch of Mathematics, inventions of the utmost importance have been derived, on which account, the subject deserves to be studied with the greatest attention. Ex. 5. It is required to convert ries. α into an infinite se a + x Ex. 6. It is required to convert into an infinite se a+b C bc b2c b3c + +, &c. a a2 d3 a1 b Ex. 8. It is required to convert into an infinite se ries. ૨૨ Ex. 9. It is required to convert into an infinite se Ex. 14. It is required to convert series. Ans. 1 1 1 1 + + + &c.= + + + +, &c. 1 1 1 1 51 125 625' 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 equation, which, as has been remarked by GARNIER, differs from an equality, in this, that the first comprehends an unknown quantity combined with certain known quantities; whereas the second takes place but between quantities that are known. Thus, the expresS ά sion a= 2 2' +, (Art. 102), according to the above remark, is called an equality; because the quantities a, s, and d, are supposed to be known. And the expression x-xd=s, (Art. 103), is called an equation, because the unknown quantity x, is combined with the given quantities d and s. Also, x-a= O 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, of the unknown quantity merely in its simplest form, after the terms of the equation have been properly arranged: Thus, x+a=b ; ax+bx=c; or+=d, &c. where ≈ de x notes 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 remunders will be equal. Thus, if x +5=12; by subtracting 5 from each side, we shall have x+5−5=12 - 5 ; but 5-5 0, and 12-5=7; hence x=7. Also, if x+ab-2x; by subtracting a from each side, we shall have and by adding 2x to each side, we shall have x+aa+x=b-2r-a+2x; but a-a=0, and -2x+2x=0; therefore x+2x=b-a, or 3x=b-a. Again, if ax-c=d, and c be added to each side, ax-c+c =d+c, or ax=d+c. Also, if 5x-7=x+12; by subtracting 2x from each side, we shall have 5x-7-2x=2x+12-2x, or 3x-7=12; subtracting -7, or, which is the same thing, adding +7 to each side of this last equation, and we shall have 31-7+7=12+7; but 7-7=0,.,3x=19. Finally, if a a+b=c-2x+d; then, by subtracting b from. each side, we shall have x−a+b—b=c—2x+d—b ; and adding a+2x to each side, it becomes xa+b-b+a+2x=c−2x+d-b+a+2x ; but a-a=0, b-b-0, and -2x+2x=0; therefore, x+2x=c+a−b+d, or 3x=c+a—b+d. : -X Cor. 1. Hence, if the signs of the terms on each side of an equation be changed, the two sides still remain equal be-cause in this change every term is transposed: Thus, if +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-x=x-b+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-a12+b-4x2; then, 4x2+3x=a+b +21.. Also, if 3x2-2+x=b-4x3-3x; then, 3x+4x3-3x2+ x=b+2. Hence also, if any quantity be found on both sides of an equation, it may be taken away from each; thus, if x+a=α +5, then x=5; if x―b=c+d-b, then x=c+d; because, by adding b to each side, we shall have x-b+b=c+d—b ̧ +b; 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 ~×2=4×2, or x=8, because, (Art. 155), ~×2=x. Also, if x 4 -3-a-b, then, by multiplying every term by 4, we shall have x-12=4a-4b. 3 Again, if 2x+1=x; then, 4x-3+2=2x; and 4x— 2x=3-2, or 2x=1. 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. X X X 2 3 Let +-+ 11; then, if every term be multiplied by 24, which is the product of all the denominators; we have 32×24+3×24+2×24=11X24; and 12x+8x+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. 146); we shall have 6x+4x+3x=132, an equation in its lowest terms. Cor. 2. Hence also, if every term on both sides have a common divisor, that common divisor may be taken away; 3x, a+6 2x+7 then, multiplying every term by 5, thus, if 5 5 5 we shall have 3x+a+6=2x+7, or x=1—ɑ. Also, if ax b. 3 7-x C +==· then multiplying by c, we shall C C C have ax-b+3=7-x, or ax+x=b+4. 188. If every term on each side of an equation be divided by the same quantity, the results will be equal: Because, by dividing every term on each side by any quantity, the value of the whole side is divided by that quantity; and, (Art. 51), if equals be divided by the same quantity, the products will be equal. Thus, if 6a2+3x=9; then, dividing by 3, 2a2+x=3. Also, if ax2+bx=acx; then, dividing every term by the ax2 bx acx common multiplier x, we shall have Ꮖ ac. + or ax+b Cor. 1. Hence, if every term on both sides have a common multiplier, that common multiplier may be taken away. Thus, if ax+ad=ab then, dividing every term by the com mon multiplier a, we shall have x+d=b. plier, or (which is the same thing) multiplying by, shall have x+b=4ax. we Cor. 2. Also, if each member of the equation have a common divisor, the equation may be reduced by dividing both sides, by that common divisor. Thus, if ax-a2x-abr-ab, or (ax-a3)x=(ax-a2)b ; then, it is evident that each side is divisible by ax-a3, x=b. whence Again, if x-a2=x+a; then, because x-a2=(x+a) .(x-a), it is evident that each side is divisible by x+a; and x2-a2 x+a 189. The unknown quantity may be disengaged from a divisor or a coefficient, by multiplying or dividing all the terms of the equation by that divisor or coefficient. |