product of the denominators, or by a common multiple of them; and if the least common multiple be used, the equation will be in its lowest terms. by 12, which is the least common multiple of 2, 3, 4; 6x + 4x + 3x = 156. COR. 2. Hence also, if every term on both sides have a common multiplier or divisor, that common multiplier or divisor may be taken away; Thus, if ax+abr=cdx; each term being divided by the common multiplier x, ax + ab = cd. COR. 3. Also, if each member of the equation have a common divisor, the equation may be reduced by dividing both sides by that common divisor; 2 Thus, if ax2-ax=abx - ab, each side is divisible by ax a2, whence x = b. COR. 4. Hence also any term of an equation may be made a square, by multiplying all the terms of the equation by the quantities necessary; as, if ax2+bcx=cď, the first term may be made a square by multiplying each term by a, and a2x2 + abcx = acd2. 2 2 (19.) If each side of an equation be raised to the same power, the results are equal; Thus, if x = 6, x2 = 36; if x + a = yb, then x2 + 2a x + a2 = y3 Qby + b2; 2 And if the same roots be extracted on each side, the results are equal : 2 2 = a a3 b3, then x = ab; if x2 + 2x + 1 = y-y+, then x+1=y-, and if x2 - 4 ax + 4a2 = y2+6 by +962, then x2 ay+3b. 2 For (13 and 14) when equal quantities on each side of an equation are multiplied or divided by equal quantities, the results will be equal. COR. Hence, if that side of the equation which contains the unknown quantity be a perfect square, cube, or other power, by extracting the square root, cube root, &c. of both sides, the equation will be reduced to one of lower dimensions : Thus, if x+8x + 16 = 36, x + 4 = 6, if x2 + 2 x3 + x2 = 100, x2 + x = 10. (20.) Any equation may be cleared of a single radical quantity by transposing all the other terms to the contrary side, and raising each side to the power denominated by the surd. If there are more than one surd, the operation must be repeated. Thus, if x = √ax + b2, by squaring each side r2 = ax + b2, which is free from surds. Also, if x2+7+x=7, then (17) by transposition, √x+7=7-x; and (19) by squaring each side, x+7=49-14x+x2, which is free from surds. Also, if x + √ax = b, 3 then (17) by transposition, a2x=b-x ; and (19) by cubing each side, a2x= b3 − 3 b2 x + 3 b x2 — x3, which is free from surds. 2 Also, if2+x2 + 21−1 = x, then (17) by transposition, √x+√x2+21 = x + 1, 2 and (19) by squaring each side, x2+ √x2+21=x2+2x+1; .. (17. Cor. 3.)√x2+21=2x+1, and (19) by squaring each side, x2 + 21 = 4x2 + 4x + 1, which is free from surds. 3 And, if a2x + √a3x3 = c, (19) by cubing each side, ax + √ a3 x3 = and (17) by transposition = c3, . (19) by squaring each side, a3x3 = c6 — 2 a2 c3 x + ax, which is free from surds. (21.) Any proportion may be converted into an equation; for the product of the extremes is equal to the product of the means. Let a b c d, by the nature of proportion .. (18. Cor. 1.) ad = bc. a = b (22.) EXAMPLES in which the preceding Rules are applied, in the Solution of Equations. 1. Given 4x + 36=5x+34, to find the value of x. (17) By transposition, 3634 = 5x — 4x, and therefore 2 = x. 7= +, to find the value of x. 5 Here 15, the product of S and 5, being their least common multiple, every term must be multiplied by it (18. Cor. 1.), and 15 x 105 3x+5x; ..(17) by transposition, 15x3x 5 x 105, or 7x = 105; and therefore (18. Cor. 2.) x = = 105 = 15. 7 3. Given 3 ax 4ab2ar-6 ac, to find the value of x in terms of b and c. .. (18. Cor. 1.) 21x39 10x+5; ..(17) by transposition, 21r-10x=39+5, and (18. Cor. 1.) 126x — 52265x — 156; (17) by transposition, 61x366; 24. Given 12-x: :: 4 1, to find the value of x. x 2 (21) Since the product of the extremes is equal to the product of the means, .. (17) by transposition, 12 = 2x + x = 3x, = 6x222x 57 3x; .. (17) by transposition, 6x-2x+3x=57-22, (18. Cor. 1.) multiplying by 10, the least common multiple of 2 and 5, 30x+4x+12=50+55 x − 185; - .. (17) by transposition, 12- 50+18555x — 30x — 4x, (18. Cor. 1.) Multiplying every term by 3, and (17) by transposition, 6x+4x-3x=18+6+4, Since 16 contains 8 and 2, a certain number of times exactly, it will be the least common multiple of 16, 8, and 2; and therefore (18. Cor. 1.) multiplying both sides of the equation by 16, |