8c2-5b +4d-9a + 7u?, or 4d--56-9a +82 + 7a, are equivalent expressions ; though it is usual, in such cases, to take them so that the leading term shall be positive. Ex. 2. 7x+y+ 30+3cy + 10ax+4a+ya + 4x2. Ex. 3. 4x3---3xy+3y --3 - 3x2 — 7x3 - 5xy+y+19+ 3x2 + 2x In Ex. 2. Collecting together like quantities, and beginning with 3x, we have 3x + 5x – --=83 --= (8-1)x=7x; 5y-y--3y =(5-1-3)y=(5-4) y=y; d+2d=(1+2)d=3d; 5xy-2xy=(5-2)xy 3xy; and 3ax+7ax=(3+7)ax=10ax; besides which there are three quantities +40, +yo, +4x2 ; ; which are unlike, and do not coalesce with any of the others; the sum required therefore is, 7x+y+3d+ 3xy+10ax +4a+y* +4x2. In Ex. 2. Beginning with 4x3, we have, 4x3 +508 - 2x3=(4+5--2)=(9-2)x3=7x3; 3xy +3xy-5xy=3xy-(5 + 3) xy=(3-8) xy= ---5xy ; 3 3 +3y+5y-7y=(3+5)y-7y=(8—7)y=+y; -3+30-8330—(8+3)=30-11=+19; 2x2 – 3x2 – 2x2 +6x9 = 8x2 – (3+2) xc=(8-5) – = 3 x2 = 3x2 ; 5y2 – 3y - 2y =5y? -- (3+2) y =(5-5) ya =0 - : . When quantities with literal coefficients are to be added together; such as mx, my, pxa, qy', &c. (where m, n, p, q, &c., may be considered as the coefficients of x, y, 2°, yo, &c.) it may be done by placing the coefficients of like quantities one after another (with their proper signs), under a vinculum, or in a parenthesis, and then, annexing the commón quantity to the sum or difference. Ex. 4. (a+b)x+(6+d)y +36 Ex. 5. (ate)x3 +(6—d)x2 +(c-f)x In Ex. 4. The sum of ax and bx, or ax-t-bx, is expressed by (a+b)x; the sum of +by and +dy, or +by+dy, is = +(6+d)y. In Ex. 5. The sum of ax3 and ex3, or ax: tex, is =(ate); the sum of +6x2 and -dx?, or +622-d, is =(-d)x2 ; and the sum of tocx ; and -fc, or 4-cu-fx, is = +(c-f). Any multi nomial may be expressed in like manner, thus ; the multinomial mx+nra -px?-90may be expressed by (mtn-p-9).x'; and the mixed multinomial pay+qyo — ray+my-nzy, by (P--r-n) cy+(9+mly: ; &c. Ex. 6. Add 2x2 + y2 +9, 7xy-3ab—*°, 4xy-y --3, and woy- xy + 3x2 together. Ans. 4x2 +yo +10xy-3ab-y+xy. Ex. 7. Add together 72a”, 24bc, 70xy, - 18a2, and -12bc. Ans. 54a2 + 12bc+70xy. Ex. 8. What is the sum of 43.xy, 7x",-12ay, -4ab,3x2, ond - 4ay? Ans. 43xy +4x9-16ay-4ab. Ex. 9. What is the sum of 7xy,—16bc, -12xy, 19bd, and 5xy ? Ans. 2bc. Ex. 10. Add together 5ax,-60bc, Tax, -4xy, --ax, and -12bc. Ans. 6ax-72bc --- 4xy. Es. 11. Add 8a r2-3ax, 70x-5xy, 9xy--5ax, and xy + 2ao xo together. Ans. 10aox?-ax+5xy. Ex. 12. Add 2x2-3y +6, 9xy—3ax-12,4ya – y-6, and xạy—3xy+3x together. Ans. 4x3 +33 +6xy—3ax-y+way. Ex. 13. Add 3x3_-423+, 5x2y—ab+x3, 4x2 . , --x?, and 2x3–3+2x3 together. . Ans. 4x__$+5x2 +5x2y—ab—X3_-3. -203 Ex. 14. Required the sum of 4x3 +7(a+b)", 4y? --5(a+b), and a3-4x2-3y:-(a+b)'. Ans. a3 +33 + y2 +(a+b). Ex. 15. Required the sum of ax*—623 +cra, bxca-acx3c2x, and axa to-bx. Ans. ax4-(+ac)x3 +(c+bcta)x+(c +b)x+c. Ex. 16. Required the sum of 5a+36—40, 2a 5b4-6cf2d, a-4b^2c +3e, and 7a +46-30~6e. Ans. 150-26-3c+2d-3e. 1 § II. Subtraction of Algebraic Quantities. 62. Subtraction in Algebra, is finding the differ. ence between two algebraic quantities, and connecting those quantities together with their proper signs : the practical rule for performing the operation is deduced from the following proposition. 63. To subtract one quantity from another, is the same thing as to add it with a contrary sign. 03, that to subtract a positive quantity, is the same as to add a negative ; and to subtract a negative, is the same as to add a positive. Thus, if 3a is to be subtracted from 8a, the result will be sa-3d, which is 5a; and if b-c is to be subtracted from a, the result will be a-(6-c), which is equal to a-b+c: For since, in this case, it is the difference between 6 and c that is to be taken from a, it is plain, from the quantity b--C, which is to be subtracted, being less than b by C, that if b be only taken away, too much will bave been deducted by the quantity c; and therefore e must be added to the result to make it correct. This will appear more evident from the following consideration; Thus, if it were required to subtract 6 from 9, the difference is properly 9--6, which is 3; and if 6-2 were subtracted from 9, it is plain, that the remainder would be greater by 2, than if 6 only were subtracted ; that is, 9-(6-2) :9–6+2=3+2=5, or 9-6+2=9-455. Also, if in the above demonstration, b-c were supposed negative, or b-c=-d; then, because c is greater than 6 by d, reciprocally c-b=d, so that to subtract -d from a, it is necessary to write n-td. 64. The preceding proposition demonstrated after the manner of Garnier. Thus, if b-c is to be subtracted from the quan tity a; we will determine the remainder in quantity and sign, according to the condition which every remainder must fulfil; that is, if one quantity be subtracted from another, the remainder added to the quantity that is subtracted, the sum will be the other quantity. Therefore, the result will be abtc, because a-bt-c+b-o=a. This method of reasoning applies with equal facility to compound quantities: in order to give an example; suppose that from 6a-36+40, we are to subtract, 5a --56+60; designating the remainder by R, we have the cquality, Rf5a --56-+-6c=6a--3b-+-4c: which will not be altered (Art. 49) by subtracting 5a, adding 5b and subtracting Cc, from each member of the equality; therefore, the result will be R=60--36 +40—50+56---66, or, by making the proper reductions, R=a+2b-2c. 65. Another demonstration of the same proposition in Laplace's manner. Thus, we can write, a=a+b-b....(1), ) 07-6--6 so that if from a we are to subtract tb or which is the same, if in a we suppress +b, or -1, the remainder, from transformation (1), must be a-6 in the first case, and a+b in the second. Also, if from a--c we take away tb or -6, the remainder, from (2), will be a------h, or a-rth. b, or |