802-5b +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+ 3d+ 3xy +10ax +4a+y +4x2. Ex. 3. 7x3-5xy+y+19+30° + 2x In Ex. 2. Collecting together like quantities, and beginning with 3x, we have 3r+5x -x=82-1= (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 + 4a, +yo, +4x”; which are unlike, and do not coalesce with any of the others; the sum required therefore is, 7x+y+-3d+3xy + 10ax +4a+ya +4x2. In Ex. 2. Beginning with 4x3, we have, 4.03 +503 - 213=(4+5-2).x'=(9-2)x3=7x3; 3xy +3xy - 5xy=3xy--(5+3)xy=(3-8)xy= ---5xy ; +3y + 5y7y=(3+5)y-7y=( 87)y=+y; --3+30---8=30-(8+3)=30-11=+19; 2x2 – 3x? - 2x2 +6x3 = 8x2 - (3+2) 2* =(8-5) x3 = +382 ; 5ya – 3y - 2y =5ye - (3+2) y =(5-5) ys=0 Xuo=0; +2x=2x. 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, 23, 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 common quantity to the sum or difference. Ex. 4. (a+b)x+(6+d)y +36 Ex. 5. (a +e)x3 +(6-d)x2 +(0-f) In Ex. 4. The sum of ax and bx, or ax+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 =(a+e)x; the sum of +6x2 and -dxa, or +622-dæ, is =(b-do; and the sum of tocx and-fc, or +.cx-fx, is = +(c-f). Any multinomial may be expressed in like manner, thus; the multinomial mxa+nx-px-9ao may be expressed by (mtn-p-9)x? ; and the mixed multinomial pxy+qyo — rxy+my-nxy, by (p--r--n) cy+(9+mly'; &c. Ex. 6. Add 2x2 + y2 +9, 7xy-3ab-22, 4xy-Y --9, and xy-- xy + 3x2 together. Ans. 4x +y +10xy-3ab-y+xy. Ex. 7. Add together 72a2, 24bc, 70xy, - 18a", and -12bc. Ans. 54a2 + 12bc+70xy. Ex. 8. What is the sum of 43xy, 7x",-12ay, -4ab,--3x, ond - 4ay? Ans. 43xy +4x3--16ay-4ab. Ex. 9. What is the sum of 7xy,—16bc, -12xy, 186d, and 5xy? Ans. 2bc. Ex. 10. Add together 5ax,-60bc, 7ax, --4xy, -Gax, and -12bc. Ans. 6ax—72bc--4xy. Ex. 11. Add 8ao x2-3ar, 7ax-5xy, 9xy---5ax, and xy + 2aa together. Ans. 10a—ax + 5xy. Ex. 12. Add 2x—3y2 +6, 9xy--3ax-92,4ya y-6, and x2y—3xy +30% together. Ans. 4x'+ya +6xy-3ax-y + x^y. Es. 13. Add 3x3_403 + x2, 5ray-ab+x3, 4x2 -«°, and 2x3−3+2x} together. Ans. 400 4x3_25+5x2 +5r4y—ab—X2—3. Ex. 14. Required the sum of 4x3 +7(a+b), 4y2 --5(a+b), and a3—4x2--3y:-(a+b). Ans. a+3+ya +(a+b). Ex. 15. Required the sum of ax*—6x3 +cra, bxca--acx-cx, and axs to-bx. Ang. ax'-(b + ac)x3 +(c+bc+a)xa +(c +b)x+c. Ex. 16. Required the sum of 5a +36-4c, 2a5b +-6c+-2d, a--4---2c+ 3e, and 7a+46-30-6e. Ans. 150-26-3c+2d-3e. § II. Subtraction of Algebraic Quantities. 6. Subtraction in Algebra, is finding the differ. cnce 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. 01, 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 positivc. Thus, if 3a is to be subtracted from 8a, the result will be 8a--3a, which is 5a; and if b-c is to be subtracted from a, the result will be a-(b-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 6 by C, that if b be only taken away, too much will have 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=9–4=5. Also, if in the above demonstration, b-c wece supposed negative, or b-c=-d; then, because c is greater than b by d, reciprocally c-b=d, so that to subtract -d from a, it is necessary to write a-tod. 64. The preceding proposition demonstrated after the manner of Garnier. Thus, if b-c is to be subtracted from the quantity 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 a--6+c, because a-6+-c+b-e=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 --5b+-6c; 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---2--2c. 65. Another demonstration of the same proposition in Laplace's manner. Thus, we can write, a=a+b-b (1), a-=-ct-b-b....(2); so that if from a we are to subtract tb or -b, or which is the same, if in a we suppress +b, or -6, the remainder, from transformation (1), must be a-b in the first case, and a t.b in the second. Also, if from a--c we take away tb or --b, the remainder, from (2), will be a--c-b, or a-th. . . |