ΣΥΣΤΗΜΑΤΑ 2...

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Transcript of ΣΥΣΤΗΜΑΤΑ 2...

  • - - 194

    .:

    : +=

    .

    (0,0) , 0+0=.

    :

    6-=3.

    6-=3 =6-3. .

    .

    0 0,5

    -3 0

    2

    2

  • - - 195

    .:

    6-=3 ,

    ,

    (, 6-3).

    :

    .

    :

    1 + 1 = 1

    2 + 2 = 2

    ( ,)

    .

    : (3 , -2)

    .

    + = 1

    4 -5 = 22

    : 3 2 = 1 , 43 5(-2) =22

    :

    o

    o

    2 2

  • - - 196

    .:

    .

    ,

    + =

    :

    ( , ) .

    .

    ( 0 , 1 ).

  • - - 197

    .:

    .

    .

    .

    .

    .

    .

  • - - 198

    .:

    1 :

    . (

    . 4

    ...).

    ,

    . .

    , .

    1. 12

    x

    x

    + =+ =+ =+ =

    + =+ =+ =+ =

  • - - 199

    .:

    .

    (3 , 2). x = 3 , y = 2.

    .

    1

    x 0 3

    y 4 1

    2

    x 0 2

    y 0 2

    2. 2 82 4 2

    x

    x

    + =+ =+ =+ =

    = = = =

    3. 4xx

    + =+ =+ =+ =

    ====

  • - - 200

    .:

    (2 , 2).

    .

    1

    x -4 0 1

    y 0 3

    15

    4

    x = 1

    yy xx (-1 , 0).

    4. 3 4 121

    x

    x

    + = + = + = + =

    = = = =

  • - - 201

    .:

    (-1 , 94

    ).

    : -3x + 4y = 12 -3(-1) + 494

    = 12 12 = 12

    x = -1 -1 = -1

    .

    2 :

    ( )

    1. 2

    2

    x

    x

    ====

    ====

  • - - 202

    .:

    . (0 , 0 ) , (2 , 4 ) .

  • - - 203

    .:

    1. : 2 23 ( 2) (5 3 ) 3 + + = + + .

    2. :

    1 + =

    2 2 =

    3. :

    2 1 1

    5 2 2 =

    41

    5 = +

    4 5 3 0 + =

    41

    5 =

    4.

    :

  • - - 204

    .:

    :

    :

    .

    .

    :

    :

    + 2 = 3

    4 -5 = -1

    : + 2 = 3 = 3 2 (1)

    4( 3 2 ) 5 = -1

    12 8 5 = -1 -8 5 = -1 12 -13 = -13 = 1

    = 1 (1) = 3 2 = 3-21 = 1

    =1 , =1 .

  • - - 205

    .:

    1 : x 1

    : x + 3y = -2 x = -3y 2 ( 1 )

    2 : ( 1 ) , x

    3y 2 . :

    -2x 5y = 2 -2 (-3y 2) 5y = 2 ( 2 )

    3 : ( 2 ) , y.

    y . :

    -2 (-3y 2) 5y = 2 6y + 4 5y = 2 y = 2 4 y = -2

    4 : y

    = -2 x . :

    x + 3y = -2 x + 3 (-2) = -2 x 6 = -2 x = 4

    , (4 , -2).

    1 : 1 .

    ,

    y.

    : 6x + 2y = -3 2y = -3 6x 6 32

    xy

    = ( 1 )

    1. 2 5 23 2

    x y

    x y

    = = = =

    + = + = + = + =

    2. 3 4 116 2 3

    x y

    x y

    = = = =

    + = + = + = + =

  • - - 206

    .:

    2 3 : ( 1 ) .

    :

    3x 4y = 11 6 33 4 112

    xx

    = 3x 2 (-6x 3) = 11 3x + 12x + 6 = 11

    15x = 11 6 15x = 5 5 115 3

    x = =

    4 : 13

    x = :

    6x + 2y = -3 16 2 33

    y + = 2 + 2y = -3 2y = -5 52

    y =

    , 1 5,3 2

    .

    x. :

    2x 3y = 7 2x = 3y + 7 3 72

    yx

    += ( 1 )

    ( 1 ) :

    -4x + 6y = -4 3 74 6 42

    yy

    + + = -2 (3y + 7) + 6y = -4 -6y 14 + 6y = -4

    -6y + 6y = 14 4 0 y = 10

    .

    .

    3. 2 3 74 6 4

    x y

    x y

    = = = =

    + = + = + = + =

  • - - 207

    .:

    y , :

    -5x + 2y = 3 2y = 5x + 3 5 32

    xy

    += ( 1 )

    ( 1 ) :

    10x 4y = -6 5 310 4 62

    xx

    + = 10x 2 (5x + 3) = -6 10x 10x 6 = -6

    0 x = 0

    x , .

    x ( 1 )

    y. (x , y) ().

    : -

    .

    .

    .

    :

    :

    + 2 = 3

    4 -5 = -1

    4. 5 2 310 4 6

    x y

    x y

    + = + = + = + =

    = = = =

  • - - 208

    .:

    .

    .

    4.

    + 2 = 3 (-4) -4 + -8 = -12

    4 -5 = -1 4 -5 = -1

    : -4 + -8 + 4 -5 = -12 1

    -13 = -13 = 1.

    = 1 .

    4 -5 = -1 4 51 = -1 4 = 5 1 4 = 4 = 1.

    =1 , =1 .

    i)

    3 2 :

    32

    2 5 1

    3 7 13

    x y

    x y

    + =

    + = 6 15 3

    6 24 26

    x y

    x y

    + = + =

    1.

    :

    i) 2 5 13 7 13

    x y

    x y

    + =+ =+ =+ = + = + = + = + =

    , ii) 3 2 255 45

    x

    x

    + = + = + = + =

    + =+ =+ =+ =

  • - - 209

    .:

    29y = 29 ,

    y = 1.

    1 y = 1 2x + 5 = 1 2x = -4

    x = -2 .

    (x , y) = (-2 , 1).

    ii)

    5 3 :

    53 3 2 25

    5 45

    x

    x

    + =

    + = 15 10 125

    15 3 135

    x

    x

    + =

    + =

    , :

    -15x + 10 + 15x + 3 = 125 + 135 10 + 3 = 260 13 = 260

    = 20.

    5x + = 45 = 20 , 5x + 20 = 45

    5x = 45 20 x = 5.

    (5 , 20).

    ( )2 2 22 = + , :

    ( ) ( )

    ( ) ( )

    2 2

    2 2 2

    4 2 3 2 1

    5 2 1 2 3

    x y x

    x y y y

    =

    + = +

    ( ) ( )( ) ( )

    2 2

    2 2 2

    4 4 4 3 4 4 1

    5 4 4 2 1 2 3

    x x y x x

    x y y y y y

    + = +

    + + = + +

    2. (((( )))) (((( ))))(((( )))) (((( ))))

    2 2

    2 2 2

    4 2 3 2 1

    5 2 1 2 3

    x y x

    x y y y

    = = = =

    + = + + = + + = + + = +

  • - - 210

    .:

    2 2

    2 2 2

    4 16 16 3 4 4 1

    5 4 4 2 1 2 3

    x x y x x

    x y y y y y

    + = +

    = + +

    2 2

    2 2 2

    4 16 3 4 4 16 1

    5 4 2 2 4 1 3

    x x y x x

    x y y y y y

    + = +

    + + = + + ( )

    12 3 151

    5 2 8

    x y

    x y

    =

    =

    .

    12 3 15 ( 2)

    5 2 8 ( 3)

    x y

    x y

    = +

    =

    24 6 30

    15 6 24

    x y

    x y

    =

    + =

    :

    -2x 6y + 15x + 6y = -30 24 -9x = -54 x = 6

    ( 1 ) x = 6 :

    -5x 2y = 8 -5 6 2y = 8 2y = -38 y = -19

    (6 , -19).

    .

    , x 0 y 0.

    .

    , 1 ax= 1

    y= , :

    + = 5

    = 1

    .

    3.

    1 15

    1 11

    x y

    x y

    + =+ =+ =+ = = = = =

  • - - 211

    .:

    , 2 = 6 = 3.

    3 + = 5 = 5 3 = 2.

    :

    1 3x= 1 2

    y=

    ( )

    13

    x = 12

    y =

    ( ) 1 1, ,3 2

    x y =

    .

    . ,

    x2 = y2 = 0 0 , :

    + = 13

    = 5

    :

    2 = 13 + 5 2 = 18 = 9

    + = 13 = 13 9 = 4. :

    x2 = 9 y2 = 4

    :