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X =

Here a + b -c=7+ 5 8 12 -8= 4, Ans.

ax + by 2. What is the value of

where a = 4; b=6; x = -3, 7?

btx Here ax + by, or 4 x 3 + 6X7 = 12 + 42=54.

ax+by 54 And 6 + X, or 6 + 3 9. Hence

6, Ans. b+2

9 ab

dB 3. What is the numeral value of +

b posing a = 9; 6 4; c= 3; d = 2;

and

8? ab

43 +

(9 X 4) Nab? =

+

(9 X 8) - 20 6

3

4 (9 X 4 X 4) Here, ab, or 9 X 4: 36, which, divided by c or 3, gives 12, the value of the first term. Then ax, or 9 X 8 72, from which subtracting door 8, there remains 64; which, divided by b, or 4, gives 16, the value of the second term. Therefore the sum of the first and second terms is 28. Then ab, or 9 X 4 X4= 144, the square root of which is 12, the value of the third term; and this subtracted from the sum of the former terms, because connected by the sign-, gives 16, the value of the whole expression.

ax

16.

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ADDITION. 38. Addition in Algebra is the method of finding the sum of several algebraic quantities, and connecting them together by their proper signs. This rule is generally divided into three cases.

Case I. To add like quantities with like signs.

Rule. Add the coefficients of the several quantities together, and to their sum prefix the common signs and annex the common letter or letters. Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5.

Ex. 6. 4ax 5az

6x + 3y 4x + 7y За 2a

2ax Sax 2x + 5y x + 'y 4a 9a

бах 14ax 8x + 8 5.3 + 8y Ex, 7. Ex. 8. Ex. 9. Ex. 10. Ex. 11. 3a Зах 2ay 26

2by2 5a бах 56

6by 6

by

2ax

86
y

8by
7ах 16ay
46 44

by2 280 -19ax

206 -+ 177 18bv

Зу

7y

a

ах

5ay 4ay 7 ay

2y

12a

34ay

a

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4y

22xy 17xy

ax

z

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Ex. 12.
Ex. 13.
Ex. 14.

Ex. 15.
3aza
a 2x2 72 4y 2a + en

Recent
Zar?
a 62%

8y

За 2
12axa
4a x2 3.30 Y

27
gax
За 522

Зу

θα 3.2
12ax
Ta za 4.2

y

4a z 16a

157 16x 17y 19a + 8z? Ex. 16. Ex. 17. Ex. 18.

Ex. 22. За Зbe bxy Ex. 19. Ex. 20. Ex. 21. 5xy 562 2bry 3z 3.0 +5.cy

2ax

14xy ба 46x 5bxy

2z x + xy

4ax y
12a
20.2
bcy 4z 2x +4cy

Зу
762
3b.cy

5x +2.cy 5ax a

5y lizy 2a

bx
6bcy
52 4.1 +3zy

7ах 2y 32a -22bx 18bcy 152 15.+-15ry 19ax— 154

59xy Ex. 23.

Ex. 24.

Ex. 25. 2r За - 46 7r?+ 3ry - 5bc

403 3a?+ 1 3r 2a 50 9p2 + 2ry

7bc 2a3 aʻ+ 17 4r 8a 76 11r'+ 5ry

4bc 5a3 2a2+ 4 9r+ 4a 66 på + 4ry

bc 3a? 7a?+ 3 5r + . 96 pod † gry-2bc a. art 10 23r +24a 316 2972 +23ry - 1960 159 -14a++35

.

— Ex. 26. Ex. 27,

Ex. 28.

Ex. 29.
4a 46
7y ба 50 30 13.x

3ry

5xy - 3x + 4ab 2y βα 6 23 10.x

4.cy Bay - 42 3ab 4y 3a 26 14 14.x 7xy

3xy - 52

5ab
2a 76
y

10 16x
. 5xy

-2.3
ху

ab Зу

8a 6 16 20x ху 4xy 294

28a 206 93—73x x—20xy 21xy-152+20ab

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CASE II. To add like quantities with unlike or different signs. RULE. Add all the positive or plus quantities into one sum, and all the negative or minus into another sum; subtract the less of these sums from the greater; to their difference prefix the sign of the greater sum, whether + or - and annex the common letter or letters. Ex. 1. Ex. 2. Ex. 3. Ex. 4.

Ex. 5. + 5a

52

+ За +9y - 6YNX За

2a

+ 3yN2 + 2a sum. 2x sum. + la sum. +3y sumn. 3yn x sum.

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Ex. 7.

- 2a

+Ila

is

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Ex. 9. Ex. 10. Ex. 6.

Ex. 8.

Зау 7 3a 2a 3r2

6r + 5ay

За?

ay + 8 -70 + 5r2 -3r + 2ay 8a2

2ay

9
8a 3a + pe

бау
10a?

Зау

11 a ta 32

2r + ay

13a + 12ay + 13

6r + 2ay +100% + 13ay 6 Note. In the 6th example, the sum of the positive or plus quantities exceeds the sum of the negative by lla; consequently the sign is + , according to the rule. In the 7th example, the sum of the positive or + (plus) quantities is less by 7a than the sum of the negative or (minus) quantities ; consequently, the sign

according to the rule. In the 9th example, the sum of the positive terms is 23a’, and the sum of the negative ones is 13a? ; their difference, therefore, is + 10a, which is the sum required. The other examples are wrought in a similar manner. If the positive and negative quantities be equal, the sum is nothing, and they are said to destroy each other. See example 7, right hand column. Ex. 11. Ex.12. Ex. 13. Ex. 14. Ex. 15. +9x 4a? βαΝα

6a + 43 62 + a + 4an x

2ax

4a + 82 +73 9a? anx + Tax

5a

22 Sanx

Зх +92 5a? βαλα + 10ax 2a + 73 Ex. 16. Ex. 17. Ex. 18.

Ex. 19.
3ab -+ 73 -2axx

6a + 20 baza + 5x+
3ab - 10x + ax x + 2a – 36 --2ax?
an ?

6x1
3ab 62 -3ac ба? 86 +3ax? 10.x

ab 2x +7apx + 4a - 26 - ax + 3x1 + 2ab + 7 -4ax 3a + 96 + ax + 1125 4ab 4.2 axx ? 8a - 2 + ax + 3.0+ 3x

3x =
Ex. 20.
Ex. 21.

Ex. 22. 6x +4x2 – 8 717—4(ata) a(a+b) – 31 (a-x)

b (1-2 8x + 1 6Xy+2(a+b) -4a(a+b) + ac)

)

Nax 5x -3.2* 9 2Ny+ (a+b) -2a(a+b) – Wia-c)

( +72-522

Ny—3(a+b) 5a(a+b) +141 (ax) 14.x 3.62 5 16xy4(a+b)

*

+10yle_x) CASE III. To add quantities when some are like and others unlike ; or when all the quantities are unlike.

RULE. Add the like quantities together, according to cases 1

+ вах

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+ 7a

ах

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and 2; connect the unlike quantities in any order, with their proper signs and coefficients prefixed.

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+ 3ab

Ex. 7.

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4y бу 7y 2y

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+ 5ax?

Ex. 1.
+ 4ab + 4

Ex. 2.
Éx. 3.

Ex. 4. 4ab + 12

3axt + 10 Nax + 3y + 4axt Tab 14 axt 3Nax

5axt ab + 3 5axt + 4Nax + 4y + 2axt 5ab 10

6axt
12 ax

2y + baxt
5
3Na-x=3axt Nax

4y + 7axt Ex. 5. Ex. 6.

Ex. 8.

Ex. 9. 5a 3ax? + 8.2 + 3y

3a? 3b*y 4a 4ax? 5.

ба?

95% ба 8ax? -16x3

10a? 106%y3 3a bax? 323 + 10a?

196*y ta

2x3
+ 14a?

26*ys
2ax? 828 + 3y

6a"

196*43 Ex. 10.

Ex. 11. 2a 3ab + 288 3a? +3a*x?—Warry} 2a + a + 363 50

-xy+5^'ab-'ab 100 + 5ab 208 + 4c -xy1263+ax? 16a2 + 20ab be 80

+4(ax): -2/3 --- cd 13a + 22ab + 368 +

4a—XYNxy+3x axt 68 + 20 -bc.

4N ab--1413xycd. Ex. 12.

Ex. 13. 54 3c? 7Nbe

ab+

3a’ +1c- é+-10 Add

+ 6

d

-5a614+ 2e_15 sa 70 12bc ab+x

4a_960—10e? +21 And

7
+

-60'+ bc- 9e +16

+ 3a

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баху

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3.c + xy
x2 + xy

3xy
6x + xy

ах

+32

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7xy

Ex. 17.
Ex. 18.

Ex.'19. Ex. 20.
бry 2ar 30

12.30 4ax 3.x2 2az

Зry

4.22
5.x?
xy
32*

2xy + 4x2
4ах

4x2
3NX+ 10

3zy + 4x2
4xy
8x2 20

14xy 822
Ex. 21.

Ex. 22. 8a?z? Зах

+ ax* +1062 3aRx 7ах 5xy

ax2 -62

+

2a®zul 9ry bax

+3ax? +50 2a’x 2a* x2 + xy

axt taʼza + 120 10ax2 + 5xy

2axt +962 + 3a*x - ax + 170

т? aʻx Ex. 23. Ex. 24. Ex. 25.

Ex. 26.
4x
+ 6xy —1222

4ax –130x + 3x3 40

5x - 4x + 3xy + Зах 9x2 2x 3x + 4.2 - 2xy

40* + 90 + ву 2y

- 3xy + 4x2 W x -+ 40 6x? 67—3x 12x—2y 4xy 822 7ax+8x?+7xy

Note. When quantities with literal coefficients are to be added together, it may be done by axto

ax: +bx placing the coefficients, with çatd cx2_dx their proper signs, under a (a+c)x+b+d (a+c)x+1-d) vinculum, or between brackets, and then subjoining the common quantity to the sum or difference thus arising, as in the margin. Ex. 1.

Ex. 2. ax+by

mtadz cdc+ady

1x--n2 bic-cy

bx+cez

dx2 (a+cd+b)x+(6+adc)y? (d+b+11)x+(adntce-m)2 Ex. 3.

Ex. 4. ax +dy

Nx+by

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mz

by dx

ax-Z

-by my

amy+NX lamd)x+(d_by: +(6+mby

dzty (am+B+1)y+(c+1)/x+(d^1)ztax

EXAMPLES FOR PRACTICE.

-6 1. Required the sum of and

Answer, a.

2 2. Add 5x^3a +6+7, and 4a-3x+26-9, together.

ath

an

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