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3. Add 3a+2145, at-b-c, and 6a-2+3, together. 4. Add 2a+36-46-9, and 5a-36+2010, together. 5. Add 2 tax'+ba+2, and x+ca? #dx-1, together.

SUBTRACTION.

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39. SUBTRACTION in Algebra is the method of finding the difference between two algebraical quantities, and connecting those quantities together with their proper signs.

RULE 1. Set those quantities from which the subtraction is to be made in one line, and the quantities to be subtracted in a line below them, observing to place like quantities under each other when they occur.

RULE 2. Subtraction in Algebra is performed by simply changing the signs (+ into and

into +) of the lower line of quantities to be subtracted, and then adding or connecting them as in addition, and the result will be the difference required. Ex. 1.

Ex. 2.
From 8a or changing the ) To 8a From + 6a + 5y

7 Take 3a sign of 3a Add-3a Take

3a + 2y

- 3 The

answer or remainder, + 5a + 9a + 3y 4 Here the quantity to be subtracted is - 3a+2y 3; we there. fore change its signs according to the 2d rule, and it becomes +3a - 2y + 3, and this added to the other quantity gives the remainder +92 + 3y – 4. Ex. 3. Ex. 4.

Ex. 5. 5a? 26 22-2y+3

5xy+ 8x^2 8y

+By – 2y 5 2a + 5b 4.x?+ 9y5

3.cy- 82—7

- y + 3y + 2 3a? - 76 -3x-lly+8

2xy+16x+5 + +9 -- Бу

5y

7 Ex. 7.

Ex. 8. Ex. 9. Ex. 10. 10 - 80 -3.cy 5xy18 5°у8) 4Nar2x°у

-5xʻy86 -* +3

ху

-xy+12 +3x*y76 Nax-5xy 777-2.cy

6xy30 —8x'yb Nax—2xy +-5%

Ex. 6.

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CX

tbc

C

Ex. 13.

Ex. 14.
bab 262 -
ctbc-b
axi 6x2

d
-2ab 62

+ bx + ex — 2d 7ab + 62 - 6

ax 262 + (cm) x + d Ex. 15. Ex. 16.

Ex. 17. -6a+132—46–12c 6x'y—31 (xy) - 4xy+12a (ar+6) -90+ 4.2+46 50 3x+y+3N (wy) 7xyể+ 4ax (ax+6). 3a + 9x_86

3x+y6N (xy) - 11xy' + 8a (ax+6) Ex. 18. From the sum of 4ax—150+42+, 52°+3ax+1024, and 90—2ax—12x (x), take the sum of 2ax–80+72*, 722-Saz

, 7x+ -70, and 30—40 (0)-2x+4a*x?.

4ax—150+ 4x and 2ax-S0+7 724
3ax+52° +100

-Sax—70+7x
-2ax +90 – 12x1

4aRx+30_4_2_27 Then, 5ax— 60+51 +22

2x -6ax—12043c+5x+4à°2 -6ax—120+504 +3x2+4a?z? Ans. llax-+60-N2-4a'r.

EXAMPLES WITH LITERAL COEFFICIENTS. Note. When the quantities that are to be subtracted have lit

ax' +63 eral coefficients, the opera

cx tion may be performed by placing the coefficients, with (a—c)2+..— (a–c)2+(6+cz their proper signs, between brackets, as in addition, and then subjoining the common quantity, the same as in the margin. Ex. 19. From pay + quz-rzo+s

Take mxy-pqxznz'ta
Remainder, (

pm)xy+(1+p) qaz+in-r)2 +s-a. Ex. 20. From a(z—y7+bxy+ca+2)2

) x
Take x,y)2 —bxy+ (a+c) (a+x).
Remainder, (a—1) (x−y)+2bxya(a +-2)».
EXAMPLES FOR PRACTICE.

ab 1. Required'the difference of and

Answer, b. 2

2 2. From 3x-2a-6+7, take &36+a+4x. 3. From 3a+b+c42d, take bm8c+2d-8. 4. From 5ab +262-c+bc-b, take 12—2ab+bc. 6. From axe-bx+cx-5d, take baé tex-12d.

ax-5
CX-d

-CX

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ato

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by 36

+46

MULTIPLICATION. 40. Multiplication in Algebra is the method of finding the product of two or more indeterminate algebraic quantities, and is generally divided into three cases.

CASE I. When both factors are simple quantities.

RULE. Multiply the coefficients of the two quantities together, and annex to the result (or product) all the letters in both factors, which will give the whole product required. If the signs of the factors be like, that is, both + or both —, the sign of the product is +; but if they are unlike, or one of them — and the other +, the sign of the product is — : and this is commonly expressed by saying, like signs give plus, and unlike signs minus.

Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. Multiply 4a

4abc 9cy -3abc -6a2bc
Зас
-2x

5a2b -26 12ab 12a+bc? -18x*y? -15a26c +12a+b%ca? Ex. 6. Ex. .7 Ex. 8. Ex. 9. Ex. 10. Ex. 11. 12a -2a

+5a -9x2 -6a'x -a'xy 36

-6x
-5bx

+52 +2xy? 36ab Sab -30ax

+45623
-30a%2° -2аory

*z Ex. 12. Ex. 13. Ex. 14. Ex. 15. Ex. 16. Ex. 17. 6Nax 12cứu -4cdx -4.x®y? 6axy? -7azy 46 -4a

-4.r'ye 3a2bą? - acéu 246x ax - 48axty +8cRdx +16xy 18a%by+23 +14a2c*r*y Ex. 18. Ex. 19. Ex. 20. Ex. 21. Ex. 22. Ex. 23. Tab —7axy -2xy +12a. 3a ́b

-.xyz 5ac +6ay

XY
-2xʻy 263a

+ ayʻz -35a2bc —42aRxy 12.3 -24a x®y 6a454

-6a.xyz Case II. When one factor is a compound and the other a simple. Rule. Multiply each or every term of the compound factor separately by the simple 'factor, and to each product prefix its proper sign, and the result will be the whole product. Ex. 24.

Ex. 25.
Ex. 26.

Ex. 27.
3a-25 6xy8

12x-ab a?—2x+1
4a
Зr

5a

4x 120?_Sab 18x*y–24.c 60ax-5a2b 4aRx-82?+4.0 Ex. 28. Ex. 29.

Ex. 30. 35x47a 12xyax+6

31xy?–4Nxta -2x 3xy

-210 -70x++14ax 362’y —3ax+y+18xy -62XY+NB8xbx-2axb

-2c

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37,

Ex. 31.

Ex. 32. Multiply 12.x®y+2xy+xy factor Aab:+3cy-abc compound. by Зах

simple 3xy or single factor. 36ax*y+6axʻy?+3axảy 12abx+y+9xcy:_3abcxy? Ex. 33. Ex. 34.

Ex. 35. 3x*—xy+2y 3y+y_2

13r_ab 522

xy

-2a 15x4—5x®y +10x*y? 3.cy +xy-2xy -26ax +2a8 36. Mult. 5ab3a-2

12a-2a'+4a-1 by 5xy

Зr 25abxy+15axy10xy 36°x-baʻx+12ax -3x7 Ex. 38. a +2b Ex. 39. 3x*—2x"+4 Ex. 40. 25xy +3a 4.22 -14ax

1322 4ax: +4623 42ax' +28ax-56ax 325x+y+39a x? Ex. 41,

Ex. 42. Multiply 9aʻx+39-11

Mult. 4.x*y+3x-2y by- 202

by 3yx -9a*x—3ax? +23-24

- 12.x*y*—9x+y+6xy CASE INI. When both factors are compound quantities.

RULE. Multiply every part of the multiplicand by each part of the multiplier, placing the products one after the other, with their proper signs; then add the several products together, as in common multiplication. Ex. 1.

Multiply a+b Ex. 2. ato Ex. 3. a'+abte by ato

ab 1st, by a..a' tab a’tab 2+ab tab? 2d, by bo. ab +62

ab 62 ab_ab2-63 Product d’+2ab+ 32 a * -62 a *

* -63 When we have two or more quantities to multiply together, it is indifferent which two we begin with; for the products will always be the same, as

will

appear from the following example. Let it be proposed to find the value or product of the four fol. lowing factors, viz: (I.) (II.) (III.)

(IV.) (a+b) (a?tab +32) (a,b) and (a'-ab+6) Ist. Multiply the factors I. and II. Next the factors III. and IV. atab+b?

a-ab+b? ato

a-6 atattate

aabtab tab-tab? +88

-abtab +2a+b + 2ab +

d2ab+2ab?—30

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a-6

.

6

*

a

It remains now to multiply the first product I. II. by the second product III. IV.

a?+2a+b+2ab? +-68
a-2a+b+2ab13
ao+2ab+2ab2+a53
-2ab-4a482_4a373_2a284
+2a+b?+4a®63+4a%b4+2ab5

-2°83_2a264—2ab5_68
*

-86 2d. Change the order of the question ; that is, multiply the factors I. and III., then II, and IV. together.

Then multiply the products I. III.,

and the II. IV. ato Then a' tab+32

at+aob? +64 ab a' - ab +62

a_72 atab a'+ab+ab?

ata*b*+a%b4 -ab-62 -ah-a?[?ab3

-a482 -26476 a * -7

ta’b? +ab8
+34 a

-66 which a* * a*b* * +64 is the product required. 3d. Again multiply the I. factor by the IV., and next the II. by the III.

It remains to multiply the

product I. IV. and II. III. a2ab+82 Next a’tab+b?

a +88 ato

ap_83 q8_a’btab? ata’b+ab?

a tao83 ta'bab? +88 -a2b-ab?—73

-Q373_78 a3 * +

a
*-63

as -86

as in the two foregoing cases. It will be proper to illustrate this example by a numerical application. Suppose a3, and b=2, we shall have a+b=5, and ab=l; further, a?=9, ab=6, and b?=4, therefore aftab+b= 19, and a'lab +6=7: so that the product required is that of 5X19X1X7=665. Now =729, and bo=64; consequently the product is a—6=665, as we have already seen. Ex. 4.

Ex. 5. *.x_15

--13576x-2424 12-15

-135-4-6x4X4

+18225—910x +2774 -3.x? +225

-810x +36x4-9-2 2452-6** +225

27242526 to 52008 Ans. 18225_1620x+23424722° +6754

ab

*

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