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ALGEBRA.

In Algebra quantities are represented by letters; the earlier letters of the alphabet, a, b, c, d, e, &c., generally standing for known quantities; and the later, x, y, z, &c., for unknown quantities. The same signs are employed as in Arithmetic, namely, + plus, for addition; -minus, for subtraction; x multiplied by, for multiplication; divided by, for division. When a number precedes a letter or symbol, it means this number multiplies the quantity after it; thus 5a = 5×a, 5ab= 5 × ab=5×a× b. In taking the sum of quantities, partly minus and partly plus, that is, with unlike signs, it is best to sum up all the positive or + quantities separately, then do the same with the negative quantities, and add the results thus separately obtained; thus, 4+3−2+3−7+8+7-8

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When a 4, b

= 8, c =

10, d=0.

1. Find the value of

5a+6b+2c+d+4a+7b+3c+5a+6b+2d

=(5x4)+(6x8)+(2 × 10)+0+(4×4)+(7×8)+ (3x10)+(5x4)+(6x8)+(2x0)

=20+48+20+0+16+56 +30 +20 +48+0=258 2. Find the value of .

6a+226 +29c+19d

= (6×4)+(22 × 8) + (29 × 10)+(19 × 0)
=24+176+290+0=490

3. Find the value of

47a+246 +16c+12d

= (47×4)+(24 x 8) + (16 x 10)+(12×0)
=188+192+160+0=540

4. Find the value of

21a+27b+51c+31d

(21 × 4)+(27x8)+(51 x 10)+(310)

84+216+510+0=810

When a = 8, b = 10, c = 12, d = 4, e = 0. 5. Find the value of

ab+5ac-2ad+7e+6de+5b

=(8×10)+5(8 x 12)-2(8×4)+(7x0)+6(4x0)

+(5×10)

=80+(5 × 96)-(2×32)+0+(6x0)+50

=610-64 = 546

6. Find the value of

4ac+15bd-7bce+6cd-4ab-7ae

=4(8×12)+15 (10 × 4)-7(10 × 12×0)+6(12×4) -4(8×10)-7(8×0)

= 384+600-0+288-320-0=952

7. Find the value of

7abd+15b-8bd-7de+14ab-8bcd+4ac

=7(8×10 × 4)+(15 x 10) -8(10×4)-7 (4×0)+ 14 (8×10)-8 (10 x 12 x 4)+4(8×12) =2240+150-320-0+1120-3840+384 3894-4160=-266

8. Find the value of

14ad-27bcd-17ae +21bd

14(8×4)-27 (10 x 12 × 4)-17(8x0)+21(10 × 4) = (14x32)-(27 × 480)—(17x0)+(21 × 40) =1288-1296011672

If a = 1, b = 2, c = 3, d = 5, e = 0.

Find the values of

(1) 2a+3b-2c+3d-4e+4a.
(2) 3a-2b+4d+4c-3d-+8.
(3) 6a-3b+4c-3d+4a-3b+7.
(4) 6c-3b+4a-2a+6b-3d.
(5) 7a+4d-3c+4a-3e.
(6) 7a-2b+4c-3d+4a.
(7) 8a-4d-4a-3b+bc-4d.
(8) 7e-6c+4b-2abc+3cde.

Ans. 21.

Ans. 24.

Ans. 2.

Ans. 11.

Ans. 22.

Ans. 4.

Ans. -36
Ans.-22.

When a quantity is squared or multiplied by itself, this is expressed by a small figure called an index placed to the right of the quantity, and a little above it. Thus 22 means 2×2; a2 means a xa, and so on.

When a number is cubed or multiplied by itself so that there are 3 terms, as 2×2×2, this is expressed by a small figure called an index placed as before, but the figure this time is 3. Thus 23 is 2 cubed, or the third power of 2; a3 is a xaxa, or the third power of a.

Similarly a means a raised to the fourth power, or axaxaxa, and so on.

When a = 4, b = 6, c = 8, d= 10, e = 0 9. Find the value of

15a2-12bc-3b2c+5c3d-d2c

= 15(4×4)

5

12 (6×8) - 3 [(6×6)×8]+
[(8x8x8) x 10]-[(10x10) x 8]
240-576-864+25600-800
- 25840—2240 = 23600

10. Find the value of

3a2-7ab2+6a2b2-d2

=3(4x4)-7[4(6×6)]+6[(4×4) (6×6)]—(10 × 10) =(3x16)-7(4×36)+6(16 × 36)-100 48-1008+3456-100 = 2396

11. Find the value of

14a2d-7a2b2+7c2d2-8e

= 14[(4×4) 10] 7 [(4×4) (6×6)] +
7[(8x8) (10x10)] - 8×0

= (14x160)-(7x576)+(7x6400)-0

=2240-4032+44800 = 43008

12. Find the value of

3abcde+4a-e-3a2ed-7de

=3(4x6x8x10x0) + 4 [(4×4) 0] —
3[(4×4)0x10] - 7 (10x0)
=0+(4×0) - (3x0)-00

13. Find the value of

a2-b2-c2-d2-e2+ab-bc+de

= (4×4)-(6×6)–(8×8)-(10 × 10)—(0x0)+

(4x6)-(6x8)+(10x0)
=16-36-64-100-0+24-48+0
40-248=
=-208

)

When a quantity is enclosed within brackets ( or [ ], it means that all the quantities are to be considered as one: thus, (8+12) means 20, (12-8) means 4, (8-12) means-4, (a+2a+c) means 3a +c, (3a-6a+c) means -3a+c, -(6a+3a—b) means -(9a-b), which is -9a+b.

Note here that when a minus sign precedes a bracket, it changes all the signs of the quantities, as in the last example; so with the sign before a bracket we can remove the bracket without affecting the equation, but if we remove the bracket having a negative sign before it, we must change all the signs.

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