19 SECTION IV. ADDITION. 37. ADDITION in Algebra is the process of finding the aggregate or sum of several quantities. For convenience, the subject is presented under three cases. CASE I. 38. When the terms are similar and have like signs. 1. Charles has 6 apples, James 4 apples, and William 5 apples; how many apples have they all? In the same way equal together to 15 a. Therefore, when the terms are similar and have like signs: RULE. Add the coefficients, and to their sum annex the common letter or letters, and prefix the common sign. 8. What is the sum of a x2, 3 ax2, 2 ax2, and 4 ax2? Ans. 10 a x2. 9. What is the sum of 3 bx, 4bx, 6bx, and bx? 10. What is the sum of 2xy, 6xy, 10xy, and 8xy? 39. When the terms are similar and have unlike signs. 1. A man earns 7 dollars one week, and the next week earns nothing and spends 4 dollars, and the next week earns 6 dollars, and the fourth week earns nothing and spends 3 dollars; how much money has he left at the end of the fourth week? If what he earns is indicated by +, then what he spends will be indicated by and the example will lars; or he earns in all 7 dollars + 6 dollars = Earning 7 dollars and then spending 4 dollars, the man would have 3 dollars left; then earning 6 dollars, he would have 9 dollars; then spending 3 dollars, he would have left 6 dol 13 dollars; and spends 7 dollars; and therefore has left the differ 4 dollars 3 dollars of 7 d, 4 d,+ 6 d, and 3 d is + 6 d. 6 dollars; hence the sum Therefore, when the terms are similar, and have unlike signs: RULE. Find the difference between the sum of the coefficients of the positive terms, and the sum of the coefficients of the negative terms, and to this difference annex the common letter or letters, and prefix the sign of the greater sum. - 10. Find the sum of — a x2, a x2, — 10 a x2, +25 a x2, + 12. Find the sum of a x3, 14 a x3, 17 a x3, — a x3, 13. Find the sum of 17 (a+b), − (a+b), (a+b), and 13 (a+b). Ans. 4 (a + b). CASE III. 40. To find the sum of any algebraic quantities." and -4b is 5 a 4b; but in The sum of 5 a and 6b is neither 11 a, nor 11b, and can only be expressed in the form of 5 a 6b, or 66 +5a; and the sum of 5 a finding the sum of 5 a, 6 b, added together by Case I., 5 a, and 4 b, the a's can be and the b's by Case II., and the two results connected by the proper sign; thus, 5 a This case includes the two preceding cases, and hence to find the sum of any algebraic quantities: RULE. Write similar terms under each other, find the sum of each column, and connect the several sums with their proper signs. (3.) 7a4b3c + 2 √√x 3b3c7√x + y 10 c +8x-y 3x 7ab10c + 6 x 4. Add together 7√x, 8x, 7x2,6√x, 4x2, - 8x, 4x, and 7 x2, -8√x. Ans. 182 12 x 7√x. 5. Add together 3 ax-4ab+ 2xy, 7ab + 5 x — 4 a, 7xy-3a4x, and abc-ax+6xy. 6. Add together 7x 3ay-5ab4c, 3 a x + 4 x +5ab5c, and 3c3ax + 7y+c. - Ans. 11x3ay + 3c + 7y. 7. Add together 5 a 32+7x+4ax-3ab, 5ab -5a224ax +4, and 6-2ab+3x+4y+ 4ax. 8. Add together 6xy+6xz - 6 m n + 4n, 4mn — 3xy+2n-8mn, - 6x z + 4n3xy +6, and 10 m n + 16 a x2, + 17 a x2 Бах. 10. Add together 17 a x2+19 a x3- 14 a x 13 a x3 5 a x1 + 6 a x2 -3 ax3 + 15 a x2. 10 a x3, and 14 a x2 Ans. 71 a 2+ 19 ax3-5 ax1. x2+19 11. Add together m+n-4a6c-7y, 8c-4m 16 bxy - - 12. Add together 8axy+17bxy 16 cxy-9axy, 18cxy+10axy-14 axz, 16 cxy + 25 axy -7bxy+25 cxy, and 10 axz + 3bxy - 10cxy + Ans. 34 axy+29bxy-3 cxy. 4 ax z. |