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. 8. Find the sum of 8 x2 y2, — 14 x2 y2, 17 x2 y2, and — x2 y2. / C - 9. Find the sum of 7 (x + y), 8 (x + y), — (x + y), and 4 (x + y). Ans. 18 (x + y). 10. Find the sum of a x2, a x2, -10 a x2, +25 a x2, and 13 a x2. 2 2 ( 11. Find the sum of 27 a b, 34 a b, 150 a b, 27 a b, and - 13 a b. Ans. - 143 a b. 12. Find the sum of ax3, 14 ax3, 17 a x3, — a x3, 44 a x3, and - а х3. 469x3 13. Find the sum of 17 (a+b), — (a+b), (a+b), and 13 (a+b). -- Ans. 4 (a+b). X X CASE III. 40. To find the sum of any algebraic quantities. The sum of 5 a and 6b is neither 11 a, nor 11b, and added together by Case I., and the b's by Case II., and the two results connected by the proper sign; thus, 5 a +6b+5a-4b10 a b. 26. 2b, x, 3y, 5x, — 3b, 3bc 3 bc. For convenience, similar terms are written under each other; then by Case I. the first column at the left is added; the second by Case II., and so on; 3 bc and -3 b c cancel. 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. 5. Add together 3 ax-4ab+2xy, 7ab + 5 x — 4a, 7xy-3a + 4x, and abc-ax+6xy. 6. Add together 7x3ay-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+ 3 x + 4 y + 4ax. 8. Add together 6xy+6xz — 6 m n + 4n, 4mn 3xy+2n-8mn, -6x z +4n3xy +6, and 10 mn 10n39. Ans. 0. 9. Add together 8am 19 nx 55bc, 19 v + 14b16cy, and 18 nx-44 am + 15 v4y. 10. Add together 17 a x2+19 ax3- 14 ax* +16 a x2, 13 а х3 5 a x2 + 6 a x2 10 a x3, and 14 a x2 + 17 a x2 -3 ax315 a x2. Ans. 71ax2+19 a x3-5 ax1. 11. Add together m+n-4a+6c-7y, 8c-4m 6n, and 12. Add together 8axy+17bxy 16 cxy-9axy, 18cxy+10axy - 14 axz, 16 cxy + 25 axy 16 bxy -7bxy 25 cxy, and 10 axz3bxy 10cxy + Ans. 34 axy+29bxy-3 cxy. 4 axz. 13. Add together 3 (x + y), 4 (x + y), and 7 (x + y). Ans. 6(x+y). 14. Add together 5 (2x+y-3 z), and 2 (2x+y - 3 z). NOTE. If several terms have a common letter or letters, the sum of their coefficients may be placed in parenthesis, and the common letter or letters annexed; thus, 6x+8x-5x=(685) x; 3b ax +36x-2 cx=(a+3b-2c) x; bexy+adxy-acxy-(bead-ac) x y. 15. Add together ax-bx+3x, and 2 ax+4bx-x. Ans. (3 a +36 + 2) x. 16. Add together by 3cy +5ay, and cy+4by -2 ay. - 17. Add together 2xy-axy, and 6xy-3axy. Ans. (84a) xy. 18. Add together 7 (3x+5y)+3a-6x+8 ab, 3 x +5(3x+5y) 7a-5ab, and 8x + 2(3x+5y) -7a-3 ab. Ans. 47x+70 y + 3 a. 41. From what has gone before, it will be seen that addition in Algebra differs from addition in Arithmetic. In Arithmetic the quantities to be added are always considered positive; while in Algebra both positive and negative quantities are introduced. In Arithmetic addition always implies augmentation; while in Algebra the sum may be numerically less than any of the quantities added; thus, the sum of 10x and 8x is 2x, which is the numerical difference of the two quantities. SECTION V. SUBTRACTION. 42. SUBTRACTION in Algebra is finding the difference between two quantities. 1. John has 6 apples and James has 2 apples; how many more has John than James? Let a represent one apple, and we have 2. During a certain day A made 9 dollars and B lost 6 dollars; what was the difference in the profits of A and B for the day? If gain is considered +, then loss and letting d represent one dol must be considered Hence it appears that, as addition does not always imply augmentation, so subtraction does not always imply diminution. Subtracting a positive quantity is equivalent to adding an equal negative quantity; and subtracting a negative quantily is equivalent to adding an equal positive quantity. Suppose I am worth $1000; it matters not whether a thief steals $400 from me, or a rogue having the authority involves me in debt $400 for a worthless article; for |