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66. Hence, we have the following general rule for the subtraction of algebraic quantities.
Change the signs of all the quantities to be subtracted into the contrary signs, or conceive them to be so changed, and then add, or connect them together, as in the several cases of addition.
EXAMPLE 1. From 18 ab subtract 14ab.
Here, changing the sign of 14ab, it becomes :--14ab, which being connected to 18ab with its proper sign, we have 18ab--14ab=(18-14)ab= 1ab. Ans.
Ex. 2. From 15x subtract -- 10.0" . Changing the sign of -10x%, it becomes +10x, which being connected to ?5x2 with its proper sign, we have 15x2 +10x2 =25?. Ans.
Ex. 3. From 24ab +7cd subtract 10ab +-7cd. Changing the signs of 18ab+7cd, we have — 18ab 7cd, therefore, 24ab+7cd - 18ab7cd=bab.
Ans. 24ab+7cd 18ab7cd
Ex. 4. Subtract 7a-55+3ax from 12a +10b+ 13ax — 3ab.
12a +10b+-13ax - 3ab Changing the signs of all the terms of 70-5b -7a+56 -- 3ax +3ar; it becomes,
c. by addition, 5a+156+100x - 3ab.
Ex. 5. From 3ab-7ax +7ab + 3ax, take
7ab +3ax Changing the signs of all ?
--4ab+3ax + 4xy the terms of 4ab- 3ax — 4xy, S
.. by addition, 6ab-ax+4xy. Ans.
In the above example, one row is set under the other, that is, the quantities to be subtracted in the lower line; then, beginning with 14a, and conceiying its sign to be changed, it becomes – 14a, which being added to 36a, we have 36a-14a=22a; also, --4b, with its sign changed, added to -12b will give 4b-126=(4-12)=-86; in like manner, 70-70 =0, and --8, with its sign changed, =+8. The following examples are performed in the same man ner as the last. Ex. 7.
Rem. <— 7a+86
Ex. 10. 7x3 +-3x2 6x3 - 2x2 +8x
Rem. 3ab---4ax +y--4cx--2x2 + 3yo, 23 45x2 - 9.0
(-a)x3+(q+c)x2 - (r+m)x+p+s)y?
67. As quantities in a parenthesis, or under a vinculum, are considered as one quantity with respect to other symbols (Art. 10,) the sign prefixed to quantities in a parenthesis affects them all; when this sign is negative, the signs of all those quantities must be changed in putting them into the parenthesis.
Thus, in (Ex. 13), when cx is subtracted from -be, the result is -6x +cx?, or --b-C).x2 ; because the sign - prefixed to (6—c) changes the signs of b and c; or it may be written +(c-b)x2.
Again, in (Ex. 14), when tmx is subtracted from --7X, the result is marx-mx; and, as this means that the sum of rx and mx is to be subtracted, that negative sum is to be expressed by ~(rx+-mx) (rt-mx. For the same reason, the multinomiat quantity --my +nya --aby-ryo +-by, when put into a parenthesis, with a negative sign prefixed, becomes
mma tab+r-6)ya. Ex. 15. From a--b, subtract a+b. Ans.--21.
Es. 16. From 7xy-5y + 3x, subtract 3xy+3y +3x.
Ans. 4xy-8y. Ex. 17. What is the difference between 7ax2 + 5xy-- 12ay +50c, and 4ax2 + 5xy-Bay--4cd.
Ans. 3ax2 - 4ay+5bc +4cd. Ex. 18. From 8x2 -- 30x + 5, take 5x2 + 2ax +-5.
Ans. 3x2 -5ax. Ex. 19. From a+b+c, take-a-b-c.
Ans. 2a-42+2c. Ex. 20. From the sum of 3x3 --4ax +3yo, 4y + Sax -3, y?-ax+5x, and 3ax—2x2 - yo ; take the sum of 5y— + x3, ax—-23 +44*, 323 —OX – 3yo, and 7y2 -ax+7.
Ans. 4x3 +-4ax -- 2y– 5x2.-7. Ex. 21. From the sum of y— oʻy--3yo, 9xy? -15—3x2y?, and 70+2x2y2 - 3x2y; subtract the sum of 5x2yz -20+xy', 3x-y-2ya tax, and 3xy? --- 4x242 —-9+a? x2.
Ans. 2xy? - 7x2y-as-a?2? +84. Ex. 22. From a 3r? yo-mox3 + 3cx-412-9; take aac?y? --no 23 +c2x+bx2 +3. Ans. (23 --2)x*yo-(ma-n*)* +(3c-ca)
XQ:-(4+b)x2 -- 12.
$ III. Multiplication of Algebraic Quantitics.
In the multiplication of algebraic quantities, the following propositions are necessary to be observed:
68. When several quantities are multiplied continually
together, the product will be the saine, in whatever order they are multiplied..
Thus, axb=b Xa=ab. For it is evident, from the nature of multiplica tion, that the product contains either of the factors as many times as the other contains an unit. Therefore, the product ab contains a as many times as ? contains an unit, that is, 6 times.
And the same quantity ab, contains b as many times as a contains an unit, that is, a times. Con. sequently, axb=barab; so that, for instance, if the numeral value of a be 12, and of b, 8, the product ab, will be 12X8, or 8 X 12, which, in either case, is 96.
In like manner it will appear that abc=cab=
69. If any number of quantities be multiplied continu
ally together, and any other number of quantities be also multiplied continually together, and then those two products be multiplied together ; the whole product thence arising will be equal to that arising from the continual multiplication of all the single quantities.
Thus, ab Xcd=axbxcxd=abcd. For ab=aXb, and cd=cXd; if x be put =cd. then ab Xcd=ab Xx=aXb Xx; but ris=cd=cxd. ...ab xarab xcxd=aXbXcd=abcd.
70. If two quantities be multiplied together, the pro
duct will be expressed by the product of their nu meral coefficients with the several letters subjoined.
Thus, 7a x5b=35ab. For 7a is =7X a, and 5b=5 Xb,..7a X 51=7 xa X5X0=7X5 XaXb=35 Xab=35ab.