tities are such as consist of different letters, or different combinations of letters; thus, 3 a, 5b; x2y, x y2 are unlike quantities. 16. In order that the student may acquire accurate conceptions of the characters and signs employed in algebra, it will be of advantage to him to compute the numerical values of the following expressions: 7. 6. bc+1 ac+2d=√48+1√54+10= √49 √64=7x8=56 (2a2+√bd-e) a-d (162+6)9-5 = 1512-5 $24 17 The term Addition, as used in Algebra, has a more extended sense than it has in Arithmetic; for here, both Addition and Subtraction, strictly so called, are employed in finding the sum. From the division of algebraic quantities into positive and negative, like and unlike, then arise three cases of addition. CASE I. To add like quantities with like signs. 18. RULE. Add together the coefficients of the several quantities, prefix the common sign, and subjoin the common letter or letters. The reason of the rule will be readily perceived by attending to the following examples: Ex. 1. a+a+a+a+a=a×5 or 5 a. This is obvious, for if a be added five times in succession, it is evidently the same thing to add 5a at once. Ex. 2.-b-26-36. The meaning of this expression is, that b is to be subtracted; from the remainder, 2b is to be subtracted; and again, from this last remainder, that 3b is to be subtracted. Now, the performing of these successive subtractions, will evidently lead to the same result, as the subtraction of 6b; and in order to indicate that 66 is subtractive, or is to be subtracted, the sign - is prefixed thus, -66. In these examples, it may be observed, that unity is understood to be coefficient of those quantities, which have none expressed. Thus, in adding up the first column of Ex. 3, we have 1+7+4+3=15. In the first column of Ex. 5, 11+5+2+1=19; and so of the rest. The following are intended as exercises to the student; the answers are not subjoined, but the correctness of the process can be proved as in addition of arithmetic. CASE II. To add like quantities mith unlike signs. 19. RULE. Collect the coefficients of the positive quantities into one sum, and those of the negative into another; subtract the less of these sums from the greater, prefix the sign of the greater, and subjoin the common letter or letters. The reason of this rule is very evident. Suppose we have to add the quantities 7a and 3a together. The meaning of this expression is, that 7a is to be added, and 3 a to be subtracted; now, to add 7a, and then to subtract 3a is manifestly the same thing as to add 4a; that is the sum of 7 a and 3a is +4a. In the same manner, if we have -7a and +3a, the sum will be 4a, because to subtract 7a, and then add 3a, amounts to the same thing as to subtract 4a. Ex. 1. Ex. 2. 4ab ax-5% 5xy+8x2. -2 -2ab-5ax +3z 7xy-7x2-9 14xy+2x2-7 5ab2ax-7z 8ab-5ax-7z Ex. 3. 2axy 8+7xy? —axy + 7— 2xy2 5axy 4+ 5xу2 -3axy-5-10xy2 3axy-10 If the sum of the positive terms be equal to that of the negative ones, their difference will of course be 0; and hence the total sum will be 0, as in the third column of Ex. 3. Ex. 5. Ex. 6. 5x3xy+4a2x 4a2y7x2 + xy 2xy2-3ax+5 Ex. 4. -4x+4xy-2a2x-2a2y+8x2+3xy 8x57xy+5a2x 5a2y+ x25xy -xy2+7ax-2 3xy2-5ax+4 2x+2xy-6a2x -3a2y-2x2-7xy -4xy2+2ax-8 CASE III. To add unlike quantities. 20. RULE. Collect the quantities into one line, prefixing their proper signs. Ex. 1. Required the sum of the quantities a, b, c and Their sum a+b+c―d. d. Ex. 2. Required the sum of the quantities 2a,-3b, ∞, 7a and 26. Their sum =2a-3b+x+7a+2b= 9ab+a. In this last example, the quantities 2a, 7a, as also -36 and 2b, can be condensed or formed into one, being like. Ex. 3. 4x2-3y+7 2xy +4y2-a Ex. 4. Sum, 4x2-3y+7+2xy+4y2—a Sum, 3ax-7ay +xy2+5a2x+ 3x2y+2a2 SUBTRACTION. 21. The term Subtraction is used in Algebra, in a sense correspondingly enlarged with that of Addition; for here both Addition and Subtraction, properly so called, are employed in finding the difference. RULE. Change the signs of the quantities to be subtracted, and then proceed as in addition. When the sign of the subtrahend is + (plus,) the reason of the rule is very evident. Thus, for example, if it were required to subtract 2 from 3a, the difference is properly represented by 3a-20, because the sign (minus,) prefixed to the 2x, shows that it is to be subtracted from 3a. That the rule leads to the true result, when any of the quantities to be subtracted have the sign-prefixed, may be shown as follows. Let it be required to subtract 2b-3x from 5 a. It is obvious that 26-3x is less than 26 by 3x, and, therefore, that if 26-3 were subtracted from 5 a, the remainder would be greater by 3x, than if 26 by itself were subtracted. Now, when 2b is subtracted, the remainder is 5a-2b, and, therefore, when 26-3x is subtracted, the remainder must be 5a-2b+3x. The same thing may be shown in the following manner: The difference of the two quantities, 5 a and 26-3x, will not be altered if the same quantity + 3x be added to both. Let this be done, then we shall have 5 a + 3x and 2b3x+3x, or 5a+3a and 2b, (because -3x and + 3x are0;) now, by subtracting 2b, the difference will be 5a+3x 2b or 5 a 2b+3a as before. It is hardly necessary to state, that the above is the result obtained by the rule. For, if the signs of the subtrahend be changed, it becomes-2b+3a, which, added to the 5 a, gives 5 a-2b+3x. Ex. 1. From 4ab-3c+4, take 2-3xy+xy2. The subtrahend with its signs changed, is -2x2+3xyxy; and, therefore, the remainder is 4 ab-3c+42x2+3xy-x y2. When any of the quantities in the subtrahend are like those in the minuend, it will be more convenient to place the one under the other, conceiving the signs of the subtrahend changed, and then proceeding as in addition. 5. From 8 ax- 4yz+7x2 take 5 a x + 3y z 8 x2 6. From 3x2 y2+4axy-7 take x2 y2-3axy+1 7. From a√ry-b√cd+xy2 take-3 a√x2 y + 2 b √ cd — 3 x y2. 22. Multiplication in Algebra, signifies the same thing as it does in Arithmetic. To multiply a by b, for example, signifies that a is to be repeated as often as there are units in b. The product is of course a xb, or simply a b, (No. 7.)* The rule for the multiplication of algebraic quantities, is commonly divided into three cases, in each of which, it must be observed, that if the signs of the factors be like, that is, either both positive +, or both negative, the sign of the product will be +; but if they be unlike, or the one positive +, and the other negative, then the sign of the duct will be -. pro *When any reference is made without stating to what, Algebra is understood; where Arithmetic is meant, it will be expressed. |