Chiziqli tenglamalar sistemasini Кramer, Gauss usullar bilan yechish

Chiziqli tenglamalar sistemasini Кramer usuli bilan echish.
Uchta noma’lumli chiziqli tenglamalar sistemasi berilgan bo’lsin:

a₁x+b₁y+c₁z=d₁

a₂x+b₂y+c₂z=d₂

a₃x+b₃y+c₃z=d₃

Asosiy va yordamchi determinantlarni tuzamiz:

a₁ b₁ c₁ d₁ b₁ c₁ a₁ d₁ c₁ a₁ b₁ d₁

= a₂ b₂ c₂ _x= d₂ b₂ c₂ _y= a₂ d₂ c₂ _z= a₂ b₂ d₂

a₃ b₃ c₃ d₃ b₃ c₃ a₃ d₃ c₃ a₃ b₃ d₃
Agar  0 bo’lsa, sistema yagona x=_x/ , y=_y/, z=_z/ yechimlarga ega bo’ladi.

Gauss usuli

Chiziqli tenglamalar sistemasini bo’lganda Кramer usuli bilan yechish qiyinlashadi. Gaussni ketma – ket noma’lumni yo’qotish usuli bilan engilroq echish mumkin.

Mavzuga doir namunaviy misollarni yechimi.
Misol-1. Ushbu determinantni yoyib yozish usuli bilan hisoblang.

4 -2 4 2 12 -10 12

= 10 2 12 =4 -2*(-2)

1 2 2 2 2 1 2


10 2

+4 =4(4-24)+2(20-12)+4(20-2)=8

1 2

Misol-2. Chiziqli tenglamalar sistemani Кramer usuli bilan yeching:

2x-3u=5

3x+2u-4z=-7

x-6y-2z=3

Yechish: Sistemaning asosiy  determinantini hisoblaymiz. Masalan, yoyib yozish usuliga asosan (Laplas teoremasi)

Demak berilgan tenglamalar sistemasi yagona echimga ega. Endi _x, _y, _z determinantlarni hisoblamiz:
5 -3 0 2 5 0

_x= -7 2 -4 = -62 , _y= 3 -7 -4 =62

3 -6 -2 1 3 -2

2 -3 5

_z= 3 2 -7 = -124

1 -6 3

Кramer formulasiga asosan sistemani yechimi x=1, y= -1, z=2 bo’ladi.

Quyidagi tenglamalar sistemasini Gauss usuli bilan yeching.

Misol-3.

Birinchi tenglamani mos ravishda 1 ga, -2 ga, -1ga ko’paytirib ikkinchi uchinchi, to’rtinchi tenglamalarga qo’shamiz.

Ikkinchi tenglamani uchunchi teglamaga va ikkinchi tenglamani -2 ga to’rtinchi tenglamani 3 ga ko’paytirib qo’shsak

Uchinchi tenglamani (-5) ga ko’paytirib, to’rtinchi tenglamani 3ga ko’paytirib qo’shsak

oxirgi tenglamadan , 3 tenglamaga qo’ysak , ikkinchi tenglamadan va birinchi tenglamadan echimi kelib chiqadi. Demak, chiziqli tenglamalar sistemasini echimi

(1; 1; -1; 1;).

Mustaqil yechish uchun misollar.
Quyidagi determinantlarni hisoblang.

1) , 2) , 3) ,

4) tenglamani yeching.

5) , 6) , 7) ,

8) , 9) , 10) ,
11) , 12) 13)
14) 15)

№ 1-20 misollarda chiziqli tenglamalar sistemasini Кramer usuli bilan yeching.

1. 2x+3y-7z=8 2. 4x-5y+z=3 3. 7x+2y-4z=-6

3x-y+5z=1 x-6y+3z=-4 4x-2y-z=-4

x-4y+6z=-7 3x+2y-z=8 -x+y+z=3

4. 5x-6y+5z=-7 5. -3x+5y+6z=9 6. -4x+8y+7z=-1

3x-2y+4z=0  5x-y-4z=-3  x+y-5z=-3

-2x+y-7z=-5 4x+3y-z=2 -2x-3y+5z=1

7. 7x+5y-3z=-4 8.  3x+4y+5z=-1 9. 3x-5y+z=0

3x-6y-5z=-1 -7x+2y-6z=3 -2x+2y-3z=1

-2x+3y+4z=3  x-y+z=-2 2x-2y+4z=-2
10.2x+7y+z=0 11. 7x-y+z=-3 12. 5x+2y-3z=10

-2x+2y-3z=1  -4x-2y-z=-3  -x-4y+z=-2

2x-2y+4z=-2  x+2y+3z=1  - -2x-y-6z=4
13.  4x+2y-z=0 14. x-3y+5z=10 15. 15 3x-2y+5z=3

 2x-y+z=5  -x-y+z=0  x+y-3z=4

 -3x+3y+2z=-2  2x+5y-3z=-4  -x-2y+3z=-3

16 . 7x+5y-3z=0 17. 4x-y+5z=3 18.  3x+2y-z=-6

8x+y+3z=-3  7x-3y-4z=5 -x-3y+2z=3

 2x+4y-4z=2  x-y+5z=-6  4x+y+z=-5

19  x+y-2z=-6 20. 2x+3y+4z=0

2x+4y-z=-1  x+y+5z=-4

 3x+5y-2z=-4  -x-2y+3z=-6

Quyidagi chiziqli tenglamalar sistemalarini Gauss usuli bilan yeching

1) 2)

3) 4)