Συστήματα γραμμικα

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συστηματα

Transcript of Συστήματα γραμμικα

  • 1 2x2.

    1 - 20

  • 1.

    2(x +1)3y = 3

    x 2(y 2) = 3x + 6

    .

    1x +

    1y =

    1

    2x +

    2y =

    2

    , 1,

    1,

    1,

    2,

    2,

    2 ! ,

    , ' . - ' .

    2x + 23y = 3x 2y + 4 = 3x = 6

    2x 3y = 12x 2y = 2

    (+)

    5y = 3 y =35

    .

    2x 3y = 1

    2x 3 35

    = 1 2x +95

    = 1 10x + 9 = 5 10x =4 x =410 x =

    25

    .

    . ( -).

    2.

    x = 13(y x)

    y = 2x + 3(y x)

    .

    x = 13(yx)

    y = 2x + 3(yx)

    x = 13y + 3xy = 2x + 3y3x

    2x 3y =1x 2y = 0

    2x 3y =12x + 4y = 0

    (+)

    y =1 .

    x 2y = 0 x 2 (1) = 0 x =2 .

    3.

    x 2y = 13x = 5(y +1)

    .

    x 2y = 13x = 5y + 5

    x 2y = 13x 5y = 5

    3x + 6y =33x 5y = 5

    (+)

    y = 2 .

    x 2y = 1 x 2 2 = 1 x = 5 .

    - 1 ( 1-20)

  • 4.

    3x +y = 52x 3y = 4

    .

    9x + 3y = 152x 3y =4

    (+)

    11x = 11 x = 1 .

    3x +y = 5 3 1+y = 5 y = 2 .

    5.

    x3

    y2

    = 3

    x 4y = 12

    .

    ,

    2x 3y = 18x 4y = 12

    2x 3y = 182x + 8y =24

    (+)

    5y =6 y =65

    .

    x 4y = 12 x 4 65

    = 12 x +245

    = 12

    5x + 24 = 60 5x = 36 x =365

    .

    6.

    x2

    y +18

    =32

    x 13

    y2

    =92

    .

    ,

    4x (y +1) = 12

    2(x 1)3y = 27

    4x y1 = 122x 23y = 27

    4x y = 132x 3y = 29

    4x y = 13

    4x + 6y =58

    (+)

    5y =45 y =9 .

    4x y = 13 4x (9) = 13 4x = 4 x = 1 .

    - 1 ( 1-20)

  • 7.

    x 3y + 2

    = 23

    y +1x + 5

    =13

    .

    y + 2 0x + 5 0

    y 2

    x 5

    .

    : ' , !!

    ,

    3(x 3) =2(y + 2)

    3(y +1) = x + 5

    3x 9 =2y 43y + 3 = x + 5

    3x + 2y = 5x + 3y = 2

    3x + 2y = 53x + 9y = 6

    (+)

    11y = 11 y = 1 .

    x + 3y = 2 x + 3 1 = 2 x = 1 .

    8.

    x +y2

    = 2x y

    3x y

    4+

    x +y3

    = 1

    .

    ,

    3(x +y) = 62(x y)

    3(x y)+ 4(x +y) = 12

    3x + 3y = 62x + 2y3x 3y + 4x + 4y = 12

    5x +y = 67x +y = 12

    5x y =67x +y = 12

    (+)

    2x = 6 x = 3 .

    5x +y = 6 5 3 +y = 6 y =9 .

    - 1 ( 1-20)

  • 9.

    (x + 3)2 + (y 8)2

    x 2 +y 2 + 39= 1

    2x + 3y + 85x + 4y 1

    =34

    .

    , !

    x2 +y 2 + 39 > 0 , x ,y ! , 5x + 4y1 0 .

    . (, ) . , .

    (x + 3)2 + (y8)2 = x 2 +y 2 + 39

    x 2 + 9 + 6x +y 2 + 6416y = x 2 +y 2 + 39 6x 16y =34 3x 8y =17 (1)

    4(2x + 3y + 8) = 3(5x + 4y1)

    8x +12y + 32 = 15x +12y3 7x = 35 x = 5 .

    (1) 3 58y =17 8y = 32 y = 4 .

    ! , x, y, ...

    .

    5x + 4y1 = 5 5 + 4 41 = 25 +161 = 40 0 .

    x, y .

    10.

    x 2(x +y) = 2y 13

    y33(x +y) = 2x 13

    .

    ,

    x 2x 2y = 2y13y9(x +y) = 6x 39

    x 4y =13y9x 9y = 6x 39

    x + 4y = 1315x + 8y = 39

    2x 8y =2615x + 8y = 39

    (+)

    13x = 13 x = 1 .

    x + 4y = 13 1+ 4y = 13 4y = 12 y = 3 .

    - 1 ( 1-20)

  • 11.

    4x y3

    = 1x2

    x +y = 3

    .

    ,

    2(4x y) = 63x 8x 2y = 63x 11x 2y = 6 .

    11x 2y = 6x +y = 3

    11x 2y = 62x + 2y = 6

    (+)

    13x = 12 x =1213

    .

    x +y = 3

    1213

    +y = 3 y = 31213

    =3912

    13 y =

    2713

    .

    12.

    x y3

    x +y

    2=

    12

    x2

    +y = 3

    .

    ,

    2(x y)3(x +y) =3

    x + 2y = 6

    2x 2y3x 3y =3x + 2y = 6

    x 5y =3x + 2y = 6

    (+)

    3y = 3 y =1 .

    x + 2y = 6 x + 2 (1) = 6 x = 8 .

    13.

    3x y = 22x + 33y 2

    = 1

    .

    3y2 0 3y 2 y 23

    .

    2x + 3 = 3y2 2x 3y =5 .

    3x y = 22x 3y =5

    9x + 3y =62x 3y =5

    (+)

    7x =11 x =117

    .

    3x y = 2 y = 3x 2 = 3 1172 =

    3372 =

    33147

    y =197

    .

    - 1 ( 1-20)

  • 14.

    x 12

    y +1

    3=

    12

    2y (3 4x) = 2(3x y)

    .

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

    2y3 + 4x = 6x 2y

    3x 32y2 = 32x 4y =3

    3x 2y = 82x 4y =3

    6x + 4y =16

    2x 4y =3

    (+)

    4x =19 x =194

    .

    2x 4y =3 2 194 4y =3

    192 4y =3

    198y =6 8y = 25 y =258

    .

    15.

    x3

    y2

    = 1

    2x 5y = 2

    .

    2x 3y = 62x 5y =2

    2x 3y = 62x + 5y = 2

    (+)

    2y = 8 y = 4 .

    2x 3y = 6 2x 3 4 = 6 2x = 18 x = 9 .

    16.

    3x y + 22

    =x + 2y

    5x 2y 3

    3=

    2x y2

    .

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

    2(x 2y3) = 3(2x y)

    15x 5y +10 = 2x + 4y2x 4y6 = 6x 3y

    13x 9y =10

    4x +y =6

    13x 9y =1036x + 9y =54

    (+)

    49x =64 x =6449

    .

    4x +y =6 4 6449

    +y =6 y =25649

    6 =256294

    49

    y =3849

    .

    - 1 ( 1-20)

  • 17.

    2x + 33y 2

    = 1

    x(2y 5)2y(x + 3) = 2x +1

    .

    3y2 0 3y 2 y 23

    .

    2x + 3 = 3y2 2x 3y =5 (1)

    2xy5x 2xy6y = 2x +1 7x + 6y =1 (2)

    (1) (2)

    2x 3y = 57x + 6y = 1

    , '

    4x 6y =107x + 6y =1

    (+)

    11x =11 x =1 .

    (2) 7 (1)+ 6y =1 6y = 6 y = 1 .

    18.

    x 1x +15

    =y 6y + 2

    x 3x

    =y 4y 1

    .

    x +15 0y + 2 0

    x 0y1 0

    x 15y 2x 0y 1

    .

    (x 1)(y + 2) = (x +15)(y6)

    (x 3)(y1) = x(y 4)

    xy + 2x y2 = xy6x +15y90xyx 3y + 3 = xy 4x

    8x 16y =883x 3y =3

    x 2y =11x y =1

    x 2y =11x +y = 1

    (+)

    y =10 y = 10 .

    x y =1 x 10 =1 x = 9 .

    - 1 ( 1-20)

  • 19.

    24x +y 5

    =1

    x + 2y +103

    4x +y 5+

    5x + 2y +10

    = 138

    .

    !

    4x +y5 0 x + 2y +10 0 .

    4x +y5 = 0 , x + 2y +10 = 0

    2

    =1

    3

    +5

    =138

    = 2 (1)

    32

    +5

    =138

    12 + 40 =1313 = 52 =4 .

    (1) = 2 (4) =8 .

    4x +y5 =8x + 2y +10 =4

    4x +y =3x + 2y =14

    8x 2y = 6x + 2y =14

    (+)

    7x =8 x =87

    .

    4x +y =3 y =4x 3 =4 873 =

    3273

    y =3221

    7 y =

    537

    .

    ( , , ).

    2 4x +y5 0 x + 2y +10 0 .

    2 1

    4x +y 5=

    1x + 2y +10

    3 1

    4x +y 5+ 5

    1x + 2y +10

    = 138

    .

    14x +y5

    = ,1

    x + 2y +10=

    2 =

    3+ 5 = 138

    3+ 5 2 = 138 13 =

    138 =

    18

    14x +y 5

    = 18

    4x +y5 =8 4x +y =3 (1)

    - 1 ( 1-20)

  • = 2 = 2 18 =

    14

    1x + 2y +10

    =14

    x + 2y +10 =4 x + 2y =14 (2)

    (1) (2)

    4x +y = 3x + 2y = 14

    , '

    8x 2y = 6x + 2y =14

    (+)

    7x =8 x =87

    .

    (1) 4 87

    +y =3 y =3327

    =2132

    7 y =

    537

    .

    x, y , .

    4x +y5 = 4 87

    5375 =

    3253357

    =567 0 .

    , x + 2y +10 = 87

    + 2 537

    +10 =87

    1067

    +10 =8106 + 70

    7=

    287

    =4 0 .

    x, y .

    20.

    x +y = 72(10x +y) = 10y + x + 25

    .

    20x + 2y = 10y + x + 25 19x 8y = 25 .

    x +y = 719x 8y = 25

    , '

    8x + 8y = 5619x 8y = 25

    (+)

    27x = 81 x = 3 .

    x +y = 7 3 +y = 7 y = 4 .

    - 1 ( 1-20)