Συστήματα γραμμικα
of 10
/10
-
Author
giwrgos-pap -
Category
Documents
-
view
15 -
download
0
Embed Size (px)
description
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)
