SECTION III. Compound Quantities. 1. Add together 3 a + 2 b and 5 x + 3 y. but In this example, compound quantities are given; they are added in the same manner as simple quantities; that is, all the terms of which they are composed are connected together with their proper signs. If we were required to add together 4 + 5 and 3 + 7, we might either say, 4+ 5+ 3 + 7 = 19; or 4+ 5 9, and 3 + 7 10, and then add these two sums, The result, both ways, is the same; = 910 19. but, in Algebra, the latter mode is not practicable, unless the quantities are similar. 2. Add together 5 b+2c, and 4 d — 3 y. ANS. 5b2c+4d-3y. This example is like the last, excepting that one of the terms is affected with the sign which must be retained in the answer; for we are not required to add the whole value of 4 d to 5 b + 2 c, but only the difference between 4 d and 3 y; and when we add 4 d, as in the answer above, we add 3 y too much, which must be subtracted. This may be rendered more intelligible, perhaps, by numbers. Add together 3+4 and 86. First, 3+4=7, and 8-62; and 7 + 2 = 9, which is the amount of the numbers given. Again, 3 + 4 +815, which is too much by 6, which must be subtracted: thus, 3+ 4+8-69, as before. 26 3. Add together 5 a + 2 c, and 3 x 4 y, and 2. ANS. 5a2c+3x-4y+2b-z. 4. Add together 5 a 2 b+d, and 3 x — 2 b, and 4 d 3 x 6 a. ANS. 5 d By adding all the terms in these several quantities, we obtain 5a2b+ d +3x-2b+4d-3x-6 a. By cancelling + 26 and -26, and + 3x and -3 x, 5 a + d + 4 d 6 d. By adding the d and 4 d, 5a5d-6 a. By balancing the + 5 a and -6 a, The answer should always be reduced, in this man ner, to the least number of terms possible. 5. What is the sum of 5 a x + 3 b c, and 7 a x 4 b c, and 3 a x + 6. Add together 3 (a 17 y? b), and 2 (a — b). ANS. 5 (a - b). In this question, 3 (a — b) — 3 a — 3 b; and 2 (ab)=2a-2b; and the answer, 5 (a—b) = 5 a-5 b. If a = 4, and b = 2, (3 a 3b) = = = (12 — 6) = 6; and (2 a — 2 b) = (8—4) = 4; and 64 10: but (5 a-5 b) = (2010) = 10 also. When the compound quantities, included in parentheses, are alike, they are added like simple quantities that are similar. The numbers or letters before the parentheses, are regarded as coefficients, and added as such; but the quantities included undergo no change. 7. What is the sum of 6 (5 a x y), and 8. Add 4 (a b-x + 3 y), 2 (a b — x + 3y), and 5 (a b -x+3y), together. 9. Required the sum of 3 a (m n-6 + y z), 6 + y z), and 4 a (m`n and 2 (b 6 + y z). x y), (bc 10. Add together 2 (b cx y), 3 4 (b c æ Y), 5 (bc — x y) From the several examples which have been given in the course of this Chapter, and the explanations with which they have been accompanied, may be derived the following general RULE for performing all questions in Addition. Connect together all the terms of the given quantities, by their proper signs, and unite such as are similar. 11. Add together 3 ab+2c, and 5 a x— c + 16, and 14 3 a x, and 5 a b + 4 c y. 12. Add together 9 x 4 y + 6 z― m n + 826+7+4b-3x + 7y-4 m n-x-8b+ 7-5 z+2y+6 x + z + 9 b — 18+ 7 m n — 6x+z+9b This example may be conveniently arranged for adding, in the following manner :— 13. Add together 5 b c + 2 ax-3y-12+ 6 a cx-8 a x + 2y - 6 b c + 18 − 9 a c x + 149bc8a x − 7 y + 177 b c 12 y— 187 a cx-6y+ax. 14.) 8x7y+ 6m+4-2x --- 15.) 8 abc-5 a (b-x) — 5+ x y z 16.) 5 a 17-7a+ 3x y 8 a +8 (x − y) — 2 x y + a (x − y) 18 8 xy +9. 18.) 9 a xy- 15+ a (a - b) — 9b c bc8a (a-b) +365 a x y 4 a (a - b)-5bca xy-13 2-4 a x y +8 b c + 5 a (a - b) CHAPTER III. SUBTRACTION. SECTION I. To subtract a Simple Quantity. 1. Ir a man receive 5 x dollars, and pay away 3 x dollars, how much has he left? ANS. 2x dollars. In this example, where the quantities are similar, we obtain the answer by subtracting the coefficients. The work is written thus, 5 x 3 x, and then the quantities are balanced; that is, the terms are reduced, according to the directions given in Addition. Let it be observed, that the sign of the quantity to be subtracted, is changed from + to — . But the sign of the quantity from which the subtraction is made, is not changed. Suppose the value of x to be 5; then 5 x = and 3 x = 15; and 25 25, 9. Subtract 5 a x from 7 a 2 of 10. Take 9 x y z from 10 x y z |