18584091 Mathimatika B Gymnasioy

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2007 41 & , .: 210 3608065, Fax: 210 3608197 www.volonaki.gr, mail: info@volonaki.gr : : :

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, 1,2 , 0,5 . : 5 , 1,2 + 0,5 5 = 3,7 8 , 1,2 + 0,5 8 = 5,2 10 , 1,2 + 0,5 10 = 6,2 , 1,2 0,5 ( ). x : = 1,2 + 0,5x x , . O , , . . . (, , , x, y, z, t, ...) , . 5 (2) + 8 : 2 3 .

17

7x + 3 + 2y + = ( + ) = ( ) . . ) 3 x + 5 x = (3 + 5) x = 8 x ) 5 + 7 = (5 1 + 7) = 11 ) y + 13 y + 5 = (1 + 13) y + 5 = 14 y + 5 , (.) . () . 7x 3y 7 x 3 y 51 (3x 2y) 51 (3 x 2 y)

1) ) 5 3 + 18 ) x + 3x 7x + 10x ) 4 + 5x + 2x ) y 3 8y + 6 2

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: ) 5 3 + 18 = (5 3 + 18) = 20 ) x + 3x 7x 10x = (1 + 3 7 + 10) x = 7x ) 4 + 5x + 2x = (4 1) + (5 + 2) x = 3 + 7x ) y 3 8y + 6 2 = (1 8) y + (3 + 6) 2 = 7y + 3 2

2) = 2 (x 2y + 1) 3(2x y 3) 4x x = 1 y = 1 : = 2 (x 2y + 1) 3 (2x y 3) 4x = 2x 4y + 2 6x + 3y + 94x = (2 6 4)x + (4 + 3)y + 11 = 8x y +11 O x = 1 y = 1 : = 8 (1) 1 + 11 = 8 1 + 11 = 18 3) = 17 [5 + 3 + ( + 9)] + = 12 = 17 (5 + 3 + ( + 9) = = 17 (5 + 3 + 9) = = 17 + 5 + 3 + 9 = = ( + ) = ( 12) = 12 4) ,

1

= 2x + x + y x y 2y : = 2x + x + y x y 2y = 2x 2y + x y + y x = 2 (x y) + (x y) + (y x) = 2 (x y) + (x y) (x y) = = = =11 =0

19

t t t t t

1) , . . 7x 4x + x 1. 5x . x + 3x 5x 2. 4x . 3 2x + 7x 3. 8x . 3x + 4x 9x 4. x 5. 7x 6. 2x 2) () () ) 15 4 + = 11 ) 2(x 1) + (x + 2) = + x ) (3 + 7) (2 + 6) = + 1 ) y + (y 2) + (3y + 2) = 3y + 2 ) 7x (x + 1) (6x + 6) = 0

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) 13x 7 + 3y x 2y + 7 (11x + y)

3. . ) = 5x 2(6 3x) + 4(2 + x) x = 3 ) = 7 ( 2) 2(3 + 3) + 5 = 7 = 1 ) = 19 2( ) (2 x) ( + 12) x = 5 = 6 4. : ) = 5( 2) 3(2 3) + 7 + = 8 ) = x + 2(3x y) 4(y 3) + x ) = 3 + (x) (y ) ( 1), x + y = 11 5. : = ( ) (2 + ) ( +) + ( 2) .

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6. = 3x 4y (5x + 7y) + (10 9x + 8y) x = 10 7. 40m. x , ) ) . 8. .x

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x. .. 3 x + 4 = 5, 8(x 2) + 3 = 5x 7, 7x 1 = 4x + 6 = =

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7x 1 = 4x + 6 1 2 x, x. 7x 1 = 4x + 6 : 7x, 4x , 1, 6 O x y, , t, z... .n

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1) : 7x + 8 = 4 3x : 7x + 8 = 4 3x 7x + 3x = 4 8 4x = 4

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x = 1 2) : 7 (x + 5) = 2(x 1) 3

23

: 7 (x + 5) = 2(x 1) 3 7x + 35 = 2x 2 3 ( ) 7x 2x = 2 3 35 ( ) 5x = 40 ( ) ( ) x = 8 3) : : =3 ( , = 3) (y + 1) ( 2y + 1) = 21 y + 1 2y 1 = 21 ( ) y 2y = 21 1 + 1 ( ) y = 21 ( ) ( ) y = 21 4) : :

24

3 (3x 1) 2 (2x 5) = 1 (5x + 1) 9x 3 4x + 10 = 5x + 5 9x 4x 5x = 5 + 3 10 0x = 2

5) : :

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10 3 5 ( + 1) = 5 ( 5 ) 30 5 5 = 25 5 5 + 5 = 25 30 + 5 0 = 0 , . 6) (x 2) + 5 = 2 (x ) . : (x 2) + 5 = 2 (x ) x 2 + 5 = 2 x 2 x 2x = 2 + 2 + 5 ( 2) x = 5 2 x , 2 = 0 = 2. 7) , 4x 3 = x . : 4x 3 = x 4x x = 3 (4 )x = 3 4 = 0 3 = 0 = 4 = 3 8) : (x 2) (3x 5) (7 + 4x) = 0

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, . : = 0 = 0 = 0 . : (x 2) (3x 5) (7 + 4x) = 0 x 2 = 0 3x 5 = 0 7 + 4x = 0 x=2 3x = 5 4x = 7

9) x + 5 = 3x 7 ) = 5, x = 6 ) x = 3, = 1 ) = +3, ) = 5 5x + 5 = 3x 7 5x 3x = 75 2x = 12

x = 6 ) x = 3, 3+5=337 3 = 9 7 5 3 = 3

= 1 ) = 3 3x + 5 = 3x 7 3x 3x = 7 5 0 x = 12 .

26

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1. : ) 13 + = 25 ) 7 = 42 ) 3 + 2 = 23 ) 256 = 187 ) 47 = 52 ) 9 + = 4 2. () (). ) 5x + 15 = 0 3 ) 2(x + 1) = 2x + 2 ) 3 (2 x) = 5 3x ) 5x 7 = 2(2x + 3) + x ) O 7 + x = 2 2 x = 7 ( ) ) x = 6 + 4x = 4 3. . ) 3 x = 6 i) 4 ) 7 x = 28 ii) 5 ) ) 4x 5 = 3x iii) 3 iv) 2 v) 10

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1. : ) 5x 7 = 8 + 2x 3 ) 2(x 3) + 9 = 5x 6 ) 9x 3(2x 5) = 21 ) 8(x 4) 6 (2 x) = 2(6x 1) 2. : ) 4(3 x) 2(3x 4) = (16 x) ) 2(3 3y) 3(1 y) = y 1

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) 3 2(3x + 1) = x 5 (5 7x) ) 6( 1) (3 + 11) = 7 3. : ) x {3 + [x (x + 3)]} = 5 ) x [ (3x + 1) 5] = 2(x + 1) ) {2(x 4) 3(x + 1) + [10 2(x + 1) 60]} = 15(x + 1) 4. : )

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11. : ) (5x 7) (3x + 12) (x 2) = 0 ) (x + 3) (2x 13) (7x + 3) = 0 ) (x 2) (x2 + 5) = 0 12. (1 x) + 3 = 2x + 5 + ) = 5, ) x = 3, ) = 2,

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13. , 7x + 5 = x + : 7x + 5 = x + :

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(1) 2=+ 2 = ) = 35, = 8 = 6cm

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2. = h + : i) h ii) 3. = + t i) t ii) 4. Q = m c , m. 5. .

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9. i) m ii) r 10. V = V0(1 + a ) . 11. F = 1,8C + 32, F C . C C F = 73,4. 12. = 2(x + y), x y . x y. : ) x 56cm y = 10cm ) T y 148cm x = 46cm.

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x = 15 15 3 15 = 45 . 3) 4 .

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:

180 . 4) O , 360 . , , . O x . 2x 3x. O 2x + x + 3x = 360 6x = 360

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x = 60 60

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40000 6000 = 16000 + 6000 = 22.000

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40000 + 3000 = 10000 + 3000 = 13.000 .

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1) 2 47. ; . 5x + 47 = 2 . 2x + 5 = 47 . 5x + 2 = 47 . 5x 2 = 47 2) 13 x . ; ; . 2(x + 1) = x 13 . x + 13 = 2(x + 1) . 2x + 13 = x + 2 . x + 13 = 2(x 1)

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