Thus, if x+9= 15, and 9 be subtracted from each side, b. And if x = a, and b be subtracted from c=d, and c be added to 7=2x+2, and 2x be taken from each side, 72; and if 2x-7=2, or 3x 7 be subtracted, or (which is the same thing) if + 7 be added to each side, 3x2 + 7 = 9. Also, if x − a+b=c-3x, then, by subtracting - a + b−3x from each side, we have x + 3x = a b + c. COR. 1. Hence, if the signs of all the terms on each side of an equation be changed, the two sides still remain equal; because in this change every term is transposed. 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. COR. 3. Hence also, if any quantity be found on both sides of an equation, it may be taken away from each; thus, if x + y = =5+y, then x = 5. If a − b = c + db, then a = c + d. (18.) 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 (13) 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. 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. If there be more fractions than one in the given equation, it may be so reduced by multiplying every term either by the 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. + + = 13; if every term be multiplied Ꮖ X 4 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 ax2 +abx = cdx; the common multiplier x, Also, if each term being divided by ax + ab = cd. Also, if (a + x2)} = 3x2. (a2 + x2), then dividing by (a2 + x2), a2 + x2 = 3x2. 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; Thus, if axa2x abx= axa2, whence x = b. ab, each side is divisible by 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 = cd', the first term may be made a square by multiplying each term by a, and a2x2 + abcx = acd'. (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 + ay- b, then x2+2ax + a2 = y2 — 2by + b2; And if the same roots be extracted on each side, the results are equal: Thus, if x2 = 49, x = 7; if x3 = a3 b3, then x = if x2+ 2x + 1 = y2 − y + 1, ab; then x + 1 = y −, and if · 4αx + 4a2 = y2 + 6by + 9b2, then x 2a = y + 3b. 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 x2+ 8x + 16 = 36, x + 4 = 6, if x3 + 3x2 + 3x + 1 = 27, x + 1 = 3, if x1 + 2x3 + 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 x2 = ax + b2, which is free from surds. Also, if √x2 + 7 + x = 7, then (17) by transposition, 2 a2 +7=7-x; and (19) by squaring each side, x2+7=49 — 14x + x2, which is free from surds. Also, if √x2 + √ x2 + 21 − 1 = x, then (17) by transposition, 2 + √ x2 + 21 = x + 1, 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. And, if vax + a3x3 = c, (19) by cubing each side, a'x + √a3x3 = c3, and (17) by transposition, √ a3x3 = c3 — a2x ; . (19) by squaring each side, a3x3 = co — 2a2c3x + a2x2, 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 C (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. 2 = X. Here 15, the product of 3 and 5, being their least common multiple, every term must be multiplied by it (18. Cor. 1.), and 15x 105 = 3x + 50; (17) by transposition, 15x-3x — 5x = 105, Given зах x in terms of b and c. (18. Cor. 2.) dividing every term by a, 3x-4b = 2x — 6c; . (17) by transposition, 3x 2x = 4b — 6c, 4. Given 3x2 10x = 8x + x2, to find the value of x. (18. Cor. 2.) dividing every term by x, 3x 10 = 8 + x; .. by transposition, 3x x8 + 10, Here 12 is the least common multiple of 2, 3, and 4; (18. Cor. 1.) multiplying both sides of the equation therefore by 12, 6x + 4x = 3x + 84; (17) by transposition, 6x + 4x — 3x = 84, (18. Cor. 1.) multiplying by 20, the least common multiple of 4 and 5, .. (17) by transposition, 5x + 120x + 4x = 1136 + 25, (18. Cor. 1.) multiplying by 6, the least common multiple of 2 and 3. |