Next, let it be required to find the sum of -3x2y, -5x2y, and -7x2y. Here x2y is taken, in the first term, -3 times; in OPERATION. the second, -5 times; and in the third, -7 times; hence, in all, it is taken -15 times. Therefore, To add together quantities having the same sign; find the sum of their coëfficients, and prefix it, with the common sign, to the literal part. -3x2y 5x1y -- 7x3y -15x2y ART. 39. SECOND CASE.-Let it be required to find the sum of +9a, -5a, +4a, and -2a. Before solving this example, the pupil must understand the following principle. Since the sign plus denotes that the quantity before which it is placed is to be added, and the sign minus, that the quantity before which it is placed is to be subtracted; therefore, the sum of two equal quantities, of which one is positive and the other negative, is zero, or 0. Thus, +a—a=0; +5a2—5a2=0; and so on. Here +9a+4a is 13a; and -5a-2a is -7a. Now, 7a will cancel +7a in the quantity +13a, and leave +6a for the aggregate, or result of the four quantities. In like manner, if it be required to obtain the sum of -9a, +5a, -4a, and +2a, we find the sum of-9a and 4a is -13a, and the sum of +5a and +2a is +7a. Now, +7a will cancel -7a in the quantity 13a; which leaves 6a for the aggregate, or sum of the quantities. Therefore, To add similar quantities having different signs; find the sum of the positive quantities and the sum of the negative quantities, then take the difference of their coëfficients, and prefix it, with the sign of the greater quantity, to the literal part. - OPERATION. +9a -5a +4a -2a +6a OPERATION. -9a +5a -4a +2a -6a ART. 40. THIRD CASE. Let it be required to find the sum of 5a2—8b+c, +b—a2, and 5b+3a2. OPERATION. 5a2-8b+c a2 + b 3a2+5b 7a2-2b+c In writing the quantities, we place those which are similar under each other, for the sake of convenience in performing the operation. We then find, as in the preceding case, that the sum of the quantities in the first column is +7a2, and in the second, -26; and there being no term similar to c, it is connected to the other quantities by its proper ART. 41. From the preceding, we derive the following GENERAL RULE FOR THE ADDITION OF ALGEBRAIC QUANTITIES.Write the quantities to be added, placing those that are similar under each other; then reduce each set of similar terms, by taking the difference of the positive and negative coefficients, and prefixing it, with the sign of the greater, to the literal part; after this, annex the other terms with their proper signs. REMARKS.-1. It is immaterial in what order the quantities are set down, if care is taken to prefix to each its proper sign. 2. It will often happen that the sum of two or more quantities is less than either. See Observations on Addition and Subtraction, page 24. EXAMPLES. 1. Add together 4ax+3by, 5ax+8by, 8ax+-6by, and 20ax+-by. Ans. 37qx-18by. Ans. 52cz-22ax2. 2. Add together 10cz-2ax2, 15cz-3ax2, 24cz-9ax2; and Scz-8ax2. 3. Find the sum of 3x2y2-10y', -x2y2+5y1, 8x2y2—6y1, and 4x2y2+244. Ans. 14x2y2-9y'. 4. Add together a+b+c+d, a+b+c—d, a+b—c+d, a—b+c +d, and Ans. 3a+3b+3c+3d. a+b+c+d. 5. Add together 3(x2—y2), 8(x2+y2), and —5 (x2—y2). Ans. 6(x2-y2). 6. Required the sum of 10a2b-12a3bc-15b2c1+10, —4a2b +8a3bc-10b2c1—4, —3a2b—3a3bc+20b2c1—3, and 2ab+12a3bc +5b2c+2. Ans. 5ab5a3bc+5. 7. Add together a2+b2+c2+d2, ab—2a2+ac—2c2+ad—2d2, a3-3ab+b3-3acc3-3ad, and 2ab-a2ac-b+2ad-c. Ans. a3+b3+c3+b2—a2—c2—d2—a—b—c. 8. Add together a"-b"+3x", 2a"-36"--x", and am-4b”—x2. Ans. 4am+2x—x2. SUBTRACTION. ART. 42. Subtraction, in Algebra, is the process of finding the simplest expression for the difference between two algebraic quantities. The quantity to be subtracted is called the Subtrahend. The quantity from which the subtraction is to be made is called the Minuend. The quantity left, after the subtraction is performed, is called the Difference, or Remainder. The explanation of the principles on which the operations depend, may be divided into two cases. 1st. Where all the terms of the quantity to be subtracted are positive. 2d. Where the quantity to be subtracted is either partly or wholly negative. ART. 43. To explain the first case, let it be required to subtract 4a from 7a. OPERATION. 7a Minuend 4a Subtrahend It is evident that 7 times any quantity, less 4 times that quantity, is equal to 3 times the quantity; therefore, 7a less 4a is equal to 3a. Hence, to find the difference between two similar quantities, we take the difference between their coëfficients, and prefix it to the common letter or letters. 3a Remainder OPERATION. a Minuend b Subtrahend a―b Remainder If it be required to subtract b from a, unless we know the number of units represented by each, we can only indicate the operation, which is done by placing the sign minus before the quantity to be subtracted. ART. 44. To explain the second case, let it be required to subtract b-c from a. OPERATION. α Minuend bc Subtrahend a-b+c Remainder If we subtract b from a, the result, a―b, is obviously too little, for the quantity b, taken from a, ought to be diminished by c before the subtraction is effected. We have, in fact, subtracted a quantity too great by c, and, therefore, to obtain a true result, the difference a-b must be increased by c; this gives, for the true remainder, a-b+c. This operation may be explained by figures, thus: Let a=9, b=5, and c=3; and let it be required to subtract 5-3 from 9. If we subtract 5 from 9, the remainder is 9-5; but the quantity to be subtracted is 3 less than 5, therefore we have subtracted 3 too much; hence, we must add 3 to 9—5, which gives 9—5+3, or 7, for the true remainder. The operation and illustration may be compared, thus: The same principle may be further illustrated by the following examples : a—(c—a)—a—c+a=2a-c. a—(a—c)—a—ate=c. a+c—(a—c)=a+c¬a+c=2c. In all these cases, we see that the same remainder would have been obtained, by changing the signs of the quantity to be subtracted, and then adding it. ART. 45. Hence we have the following RULE FOR THE SUBTRACTION OF ALGEBRAIC QUANTITIES.-Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other. Conceive the signs of all the terms of the subtrahend to be changed, from+to, or from to +, and then reduce the result to its simplest form. REMARKS.-1. Beginners may solve a few examples by actually changing the signs of the subtrahend. After this, it is better merely to conceive the signs to be changed; that if it becomes necessary to refer to the operation, we may be in no doubt with regard to the signs of the terms originally. 2. Subtraction in Algebra may be proved in the same manner as in Arithmetic, by adding together the remainder and the subtrahend; the sum should equal the minuend. 5. From 9x2-4y+9 take 7x2+5y—14. Ans. 2x2-9y+23. 6. From 23xy2-7y+11x2 take 11xy2—5y—9x2. 7. From 12x+18 take 12x-18+y. Ans. 12xy2-2y+20x2 Ans. 36-y. Ans. 4-3x2. Ans. (4a-3)x-(2+)x+c-5. 10. From-17x3+9ax2—7a2x+15a3 take—19x3+9ax2-9a2x Ans. 2x3+2a2x—2a3. +17a'. 11. From +3x2+3x+1 take x3-3x2+3x-1. Ans. 6x2+2. 12. From 9ax2-13+20 ab3x-46"cx2 take 3bcx2+9ax2-6 +3ab3x. Ans. 17 ab3x-7bmcx2-7. 13. From a-x-(x—2a) +2a—x take a-2x-(2a—x)+(x -2a). 14. From 4am+2x2-x1 take aTM-b"+3x and 2aTM—3b”—x2. Ans. am+4b"-X2. Ans. 8a-3x. REMARK.-The number of exercises in both Addition and Subtraction is purposely small, as ample practice of the best kind will be found in the operations of Multiplication and Division. THE BRACKET, OR VINCULUM. As the Bracket, or Vinculum, is frequently employed, it is proper that the pupil should become acquainted with the rules which govern its use in relation to Addition and Subtraction. ART. 46. 1st. Where the sign plus precedes a vinculum, it may be omitted without affecting the expression. This principle is selfevident. Thus, a+(b—c) is the same as a+b—c. The first shows that b is to be diminished by the number of units in c, and the remainder added to a; the second shows that a is to be increased by the number of units in b, and the result diminished by the number of units in c. Or, if a=6, b=5, and c=3, Then 6+(5-3)= 6+2=8; And 6+5—3 =11—3—8. From this it follows, that any number of terms of an algebraic expression may be included within a vinculum, if it be preceded by the sign plus. Thus, xy-2-x+(y-z). 2d. Where the sign minus precedes a vinculum, it may be omitted if the signs of all the terms within it be changed. This is evident, because the sign minus indicates subtraction, which is effected by changing the signs of all the terms of the quantity to be subtracted. Thus, Sometimes several brackets, or vinculums, are employed in the same expression; by this principle they may all be removed. Thus, |