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66. llence, 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 18ab subtract 14ab.

Here, changing the sign of 14ab, it becomes :-14ab, which being connected to 18ab with its proper sign, we have 18ah-14ab=(16-14)ab= 1ab. Ans.

Ex. 2. From 15x” subtract -10r. Changing the sign of --10x9, it becomes + 10x, which being connected to 15x2 with its proper sign, we have 15x2 +10x? =25x”. Ans.

Ex. 3. From 24ab+7cd subtract 13ab +7cd. Changing the signs of 18ab+7cd, we have --13ab -7cd, therefore, 24ab+7cd - 18ab-7cd=6ab.

Ans. Or 24ab+7cd


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Ex. 4. Subtract 7a-5b + 3ax from 12a +10b+ 13ax = 3ab.

12a +10b+-13ax - 3ab Changing the signs of all the terms of 7a ---56 -7a+ 5b -- 3a. +3ax; it becomes,

... by addition, 5a +156+10ux-3ab.

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Ex. 5. From 3ab-7ax + ab + 3ax, take


7ab +3ax Changing the signs of all

-4ab+3ax +4xy the terms of 4ab- 3ax — 4xy, S

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.. by addition, 6ab-ax+4xy. Ans.

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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 conceiy-
ing 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
46--126=(4-12)=-36; 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.

Ex. 8.
From 3x-4a+ b
Take 2x +3a-74


at 6

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Rem. 3ab- 4ax +y--4cx--2002 + 3yo, 23+5x2_90

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(6-a)x3+(9+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 sigo 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

. ) - , the result is -bx+cx?, or (b-c. 2; because the sign prefixed to (b-c) changes the signs of b and c; or it may be written +(c-b)x?,

Again, in (Ex. 14), when +mx is subtracted from ---78, the result is mr-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)=(r-mx. For the same reason, the multinomiat quantity --my +noy--aby---ryo +-by, when put into a parenthesis, with a negative sign prefixed, becomes

--(m--+abtr-6)y?. Ex. 15. From a--b, subtract a+b. Ans. -20.

Es. 16. From 7xy-5y +3.0, subtract 3xy+3y +-3x.

Ans. 4xy-by. Ex. 17. What is the difference between 7ara + 5xy--12ay +50c, and 4ax? +-5xy--Bay-4cd.

Ans. 3ax 2 -- 4ay+5bc-+-4cd. Ex. 18. From 8x2 -3ax +5, take 5x2 + 2ax +5.

Ans. 372 5ax. Ex. 19. From a+b+c, take-a-b-c.

Ans. 2a + 2+2c. Ex. 20. From the sum of 3x3 -- 4ax +3y2, 4y +

-23, y? —ax +- 5x9, and 3ax-2x2 - yo ; take the sum of 5yo — +23, ax—-03 +42°, 3.23.

5- ---ax3y”, and 7ya--ax+7.

Ans. 4x3 + 4ax -- 2y2 - 5x2-7. Ex. 21. From the sum of mayo oy--3ayo, 9xy-15-3.xoy, and 70+ 2xoy - 3xoy; subtract the sum of 5x2y3 -20+xy', 3x-y ---waya +ax, and 3xy? --4x2y? -9+a? x2.

Ans. 2xy? -7xy-ax-a?x+S4. Ex. 22. From asxa yomox3 +3cx-423-9,

-max take aa xoy --nox3 +cax + bx2 +3. Ans. (122-a2)x*yjo (m2-na).: +(30-02)

XX---(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, a Xb=b Xa=ab. For it is evident, from the nature of multiplication, 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 12 X 8, or 8 X 12, which, in either case, is 96.

In like manner it will appear that abc=cab= oca, &c.

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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 X x; but x is=cd=cxd. ...ab Xx=ab xcxd=axbxcd=abcd.

70. If two quantities be multiplied together, the pro

duct will be expressed by the product of their nu meral coefficients wilh the several letters subjoined.

Thus, 7a X 5b=35ab. For 7a is =7X a, and 5b=5 Xb, ...7a X 5627 XQ X5 X67X5 XaXb=85 Xab=35ab.

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