74. From what has been stated regarding the management of surd quantities, we can readily reduce a fraction which has a surd quantity in its denominator, to an equivalent fraction having a rational denominator. Thus, if both a NI numerator and denominator of the fraction be multia√ π plied by, we obtain the equivalent fraction a√, having a rational denominator. In the same man nator both multiplied by (a+x), it will become b(a+x) a+x In such examples, it is evident that the index of the multiplier must just be such a fraction, as that when added to the index of the given denominator, it shall produce the least possible whole number. EQUATIONS. 75. When one algebraic quantity is connected with another by means of the sign of equality, (=), the whole expression constitutes what is called an equation. Thus, ax+b=c is an equation. In equations, as applied to the solution of problems, the object is to discover the value of certain unknown quantities, from having their relation specified or given to such as are known. When the value of the unknown quantity is found in terms of those which are given, the equation is said to be solved. 76. If an equation contain only the first power of the unknown quantity, it is called a simple equation. Thus, 3x+ab is a simple equation. If it contain the second power of the unknown quantity, or the product of two unknown quantities, it is called a quadratic equation. Thus, x2+b=c; xy-ab are quadratic equations. If it contain the third power of the unknown quantity, it is called a cubic equation. Or, in general, if it contain the nth power of the unknown quantity, it is said to be an equation of n dimensions. 77. In solving an equation, the great object is to disengage the unknown quantity from the given quantities with which it is combined, so as to make it stand on one side of the equation by itself, and thus determine its value. The disengagement of the unknown quantity, according to the particular manner of its combination, may be effected on one or more of the following axioms, or self-evident principles: Ax. 1. If, to equal quantities, equals be added, the wholes. will be equal. Ax. 2. If, from equal quantities, equals be taken, the remainders will be equal. Ax. 3. If equal quantities be multiplied by equals, the products will be equal. Ax. 4. If equal quantities be divided by equals, the quotients will be equal. Ax. 5. If two quantities be equal, any like powers or roots of them will also be equal. 78. On the solution of simple equations containing only one unknown quantity. Ex. 1. Let x+5=12. If 5 be taken from each, then (Ax. 2.) x+5—5=12 — 5; but 5. 50, hence x = 12 5. Ex. 2. Let x-7=9. If 7 be added to each, then (Ax. 1.) x-7+7=9+7; but -7+7=0 0, hence x=9+7. Ex. 3. Let 3x-6=2x+2. If 6 be added to each, then 3x-6+6=2x+2+6. Again, if 2a be subtracted from each, then 3x-6+6-2x=2x+2+6-2x; but -6+6-0, and 2x 2x also = 0, hence 3x 2x=2+6. On reviewing the three preceding examples, it appears, 1st, That x+5=12, is identical with 2d, x-7=9, 3d, - 3x-6=2x+2, 3x-2x x=12-5. x = 9+7. 2 + 6. Hence it follows, that any quantity may be transferred from one side of the equation to the other by changing its sign. In the first of the above examples, it is manifest that x=7; in the second = 16; and, in the third, x=8. 79. From the conclusion now drawn regarding the transposition of quantities, it readily follows, 1st, That, if the same quantity stand on both sides of the equation, with the same sign, it may be left out of both. Thus, if x+a= b+a, then x=b+a—a, or x=b; and, 2d, That the signs of all the terms may be changed without, in any respect, changing the value of the unknown quantity. Thus, if x-a-b-c, then, by transposition, x=a+b-c. If the signs of all the terms be changed, then we shall have a-x=c-b; and hence, by transposing to the right hand side of the equation, and c-b to the left, we find ac+b=x, where the value of x is the same as before. α 80. If the unknown quantity has a coefficient, it may be taken away by dividing both sides of the equation by it. Ex. 2. 3x-16 Let 5x+8=3x+16. Then, by transposition, 5x8, or 2x = 8, and hence x = 4. Ex. 3. Let 3ax+b=cx+d. Then 3ax-cx-d-b, but 3ax cx (3a-c)x, (No. 27.); hence, (3a-c)x=d-b, and, dividing both sides of the equation by 3a-c, we have d-b 81. An equation may be cleared of fractions by multiplying each side of the equation by the denominators of the fractions in succession, or by multiplying each side by their least common multiple. Before giving an example, it may be necessary to remark, that the product of a fraction, multiplied by its de nominator, is equal to the numerator. Thus, let α b be any 2x by 2, we have += 20. Again, let each side be multiplied 3 by 3, and we have 3x+2x=60, that is, 5x=60, and hence 60 x= =12. Or, multiplying by 6, the least common mul 5 a2bd+bdx 3 a2b+b x 3 ; again, multiply by d, we have a dx+bcx= ; lastly, multiply by 3, we have 3ada+3bcx= abd+bdx; now, by transposition, 3ada+3bcx-bdx-a2bd; but 3adx+3bcx-bdx (3ad+3bc-bd)x, (No. 27. ;) hence, (3ad+3bcbd)x = a2bd, and, dividing both sides of the equaa2bd 3ad+3bc-bd tion by 3ad3bc-bd, we have x = 82. If the unknown quantity, in any equation, be in the form of a surd, it may be reduced to the rational form by transposing the terms, so that this surd quantity may stand on one side of the equation, and the remaining terms on the other, and then involving both sides to such a power as corresponds with the index of the surd. 8 Ex. 1. Let √x+3=8. Then, by transposition, = 35, now, by squaring both, we have x = 25, (Ax. 5.) |