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Άλγεβρα Β΄ Λυκείου Συστήματα ΚΑΤΗΓΟΡΙΑ 1 Γραμμικά συστήματα 2x2. Ασκήσεις 1 - 20
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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)