ALGEBRA. Introduction. THE operations explained in Arithmetic are sufficient for the solution of all questions in numbers, that ever occur; but it is to be observed, that in every question there are two distinct things to be attended to; first, to discover, by a course of reasoning, what operations are necessary; and, secondly, to perform those operations. The first of these, to a certain extent, is more easily learnt than the second; but, after the method of performing the operations is understood, all the difficulty in solving abstruse and complicated questions consists in discovering how the operations are to be applied. It is often difficult, and sometimes absolutely impossible to discover, by the ordinary modes of reasoning, how the fundamental operations are to be applied to the solution of questions. It is our purpose, in this treatise, to show how this difficulty may be obviated. It has been shown in Arithmetic, that ordinary calculations are very much facilitated by a set of arbitrary signs, called figures; it will now be shown that the reasoning, previous to calculation, may receive as great assistance from another set of arbitrary signs. Some of the signs have already been explained in Arithmetic; they will here be briefly recapitulated. (=) Two horizontal lines are used to express the words" are equal to," or any other similar expression. (+) A cross, one line being horizontal and the other perpendicular, signifies "added to." It may be read and, more, plus, or any similar expression; thus, 7+5=12, is read 7 and 5 are 12, or 5 added to 7 is equal to 12, or 7 plus 5 is equal to 12. Plus is a Latin word signifying more. (—) A horizontal line, signifies subtracted from. It is sometimes read less or minus. Minus is Latin, signifying less. Thus, 14 Thus, 1468, is read 6 subtracted from 14, or 14 less 6, or 14 minus 6 is equal to 8. Observe the signs and affect the numbers which + they stand immediately before, and no others. Thus 146816; and 14+8 — 6 = 16; and 86+14= 16 and, in fine, −6 +8 + 14 = 16. In all these cases the 6 only is to be subtracted, and it is the same, whether it be first subtracted from one of the numbers, and then the rest be added, or whether all the others be added and that be subtracted at last. (x) (.) An inclined cross, or a point, is used to express multiplication; thus, 5 × 3 = 15, or 5.3 = 15. (÷) A horizontal line, with a point above and another below, is used to express division. Thus, 15 ÷ 3 = 5, is read 15 divided by 3 is equal to 5. But division is more frequently expressed in the form of a fraction (Arith. Art. XVI. Part II.), the divisor being made the denominator, and the dividend the numerator. = Thus 5, is read 15 divided by 8 is equal to 5, or one third of 15 is 5, or 15 contains 3, 5 times. Example. 6x9 +15-37.8 — 1 + 14. This is read, 9 times 6 and 15 less 3 is equal to 8 times. 7 less 16 divided by 4, and 14. To find the value of each side; 9 times 6 are 54 and 15 are 69, less 3 are 66. Then 8 times 7 are 56, less 16 divided by 4, or 4 is 52, and 14 more are 66. In questions proposed for solution, it is always required to find one or more quantities which are unknown; these, when found, are the answer to the question. It will be found extremely useful to have signs to express these unknown quantities, because it will enable us to keep the object more steadily and distinctly in view. We shall also be able to represent certain operations upon them by the aid of signs, which will greatly assist us in arriving at the result. Algebraic signs are in fact nothing else than an abridgment of common language, by which a long process of reasoning is presented at once in a single view. The signs generally used to express the unknown quantities above mentioned are some of the last letters of the alphabet, as x, y, z, &c. I. 1. Two men, A and B, trade in company, and gain 267 dollars, of which B has twice as much as A. What is the share of each? In this example the unknown quantities are the particular share of A and B. Let a represent the number of dollars in A's share, then 2x will represent the number of dollars in B's share. Now these added together must make the number of dollars in both their shares, that is, 267 dollars. x+2x=267 Putting all the x's together, 3x=267 If 3 x are 267, 1 x is of 267 in the same manner as if 3 oxen were worth $267, 1 ox would be worth x= 89 A's share. 2 x 178 B's share. of it. 2. Four men, A, B, C, and D, found a purse of money containing $325, but not agreeing about the division of it, each took as much as he could get; A got a certain sum, B got 5 times as much; C, 7 times as much; and D, as much as B and C both. How many dollars did each get? Let a represent the number of dollars that A got, then B got 5x, C 7x, and D (5x+7x) = 12 x. These, added together, must make $325, the whole number to be divided. x+5x+7x + 12 x = 325 Putting all the x's together, 25x=325 In the same manner perform the following examples. 3. Said A to B, my horse and saddle together are worth $130, but the horse is worth 9 times as much as the saddle. What is the value of each ? Ans. Saddle $13, horse $117. |