Chiziqli tenglamalar sistemasini yechish
![Chiziqli tenglamalar sistemasini yechishning oddiy usuli solve buyrug’I orqali bajarilad.
Buning uchun maple matematik paketiga kirib solve buyrug’ini yozib ()qavs ichiga {}
qavs ochamiz va berilgan tenglamalar sistemasini vergul bilan ajratib kiritamizva {}qavs
tashqarisiga vergul quyib yechimni qaysi nomalumga nisbatan yechishini kiritamiz kata
[]qavs bilan va qatorni ; yopib inter buyrug’ini beramiz va javobni qabul qilamiz.
Gaus usuli bilan chiziqli tenglamalar sistemasini yechish uchun kutub honadan
foydalanishga to’g’ri keladi bizga kerakli kutubhonani quyidagicha chaqiramiz
>with(LinearAlgebra): va sistemaning asosiy matritsasini quyidagicha kiritamiz
A := <<1,2,3,4>|<2,1,2,3>|<3,2,1,2>|<-4,3,2,1>>;
Yordamch matritsasini esa > b := <5,1,1,-5>; ko’rinishda kiritamiz va Gaus usilidda
ishlash uchun quyidagi buyruqlarni ketma ket kiritamiz GaussianElimination(A); >
GaussianElimination(A,'method'='FractionFree'); ReducedRowEchelonForm(<A|b>);
Natija toyyor.
Kramer usulida ishlash uchun ha with(LinearAlgebra): kutub honasidan
foydalanamiz assos matritsasini yuqoridagidek kiritaniz va Determinant(y) ushbu
buyruq orqali determinantini hisoblaymiz yordamchi matritsani ketma ket ustunlar
o’rniga almashtirib ulRNING HAM determinantini hisoblab olamiz va yakuniy natijani
mos ravishda
> x1:=Determinant(y1)/Determinant(y);
> x2:=Determinant(y2)/Determinant(y);
> x3:=Determinant(y3)/Determinant(y);
> x4:=Determinant(y4)/Determinant(y);
Yechimlarni qabul qilib olamiz
Teskari matritsa usulida yechish uchun ham yuqoridagi kutubhonadan foydalanamiz
va asosmatritsasini topamiz so’ngunga teskari matritsasini topamiz u quyidagicha](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_2.png?v=1)
![topiladi A^(-1); va topilgan teskari matritsaga yordamchi matritsani ko’paytiramiz A^(-
1).B (.) buyerda matritsa ko’paytmasi amali chiqgan natija bizning Sistema yechimi.
Mavzuga doir misollar
Tenglamalar sistemasini Oddiy usul bilan yechish
> solve({x+2*y+3*z-4*b=5,2*x+y+2*z+3*b=1,3*x+2*y+z+2*b=1,4*x+3*y+2*z+b=-5},
[x,y,z,b]);[ ] [ ] , , , x -50 y 62 z -3 b 15
Tenglamalar sistemasini Gauss metodi orqali yechish:
> with(LinearAlgebra):
A := <<1,2,3,4>|<2,1,2,3>|<3,2,1,2>|<-4,3,2,1>>;
:= A
1 2 3 -4
2 1 2 3
3 2 1 2
4 3 2 1
> b := <5,1,1,-5>;
:= b
5
1
1
-5
> GaussianElimination(A);
1 2 3 -4
0 -3 -4 11
0 0 -8
3 -2
3
0 0 0 -1
2](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_3.png?v=1)
![> GaussianElimination(A,'method'='FractionFree');
1 2 3 -4
0 -3 -4 11
0 0 8 2
0 0 0 -4
> ReducedRowEchelonForm(<A|b>);
1 0 0 0 -50
0 1 0 0 62
0 0 1 0 -3
0 0 0 1 15
Tenglamalar sistemasini Kramel usulida yechish:
> with(Student[LinearAlgebra]):
> y:= <<1,2,3,4>|<2,1,2,3>|<3,2,1,2>|<-4,3,2,1>>;
> y1 := < <5,1,1,-5>|<2,1,2,3>|<3,2,1,2>|<- 4,3,2,1>>;
:= y1
5 2 3 -4
1 1 2 3
1 2 1 2
-5 3 2 1
> y2:= <<1,2,3,4>|<5,1,1,-5>|<3,2,1,2>|<-4,3,2,1>>;
:= y2
1 5 3 -4
2 1 2 3
3 1 1 2
4 -5 2 1
> y3:= <<1,2,3,4>|<2,1,2,3>|<5,1,1,-5>|<-4,3,2,1>>; := y
1 2 3 -4
2 1 2 3
3 2 1 2
4 3 2 1](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_4.png?v=1)
![:= y3
1 2 5 -4
2 1 1 3
3 2 1 2
4 3 -5 1
> y4 := <<1,2,3,4>|<2,1,2,3>|<3,2,1,2>|<5,1,1,-5>>;
:= y4
1 2 3 5
2 1 2 1
3 2 1 1
4 3 2 -5
> x1:=Determinant(y1)/Determinant(y); x1 -50
> x2:=Determinant(y2)/Determinant(y);
x2 62
> x3:=Determinant(y3)/Determinant(y);
x3 -3
> x4:=Determinant(y4)/Determinant(y);
Tenglamalarsistemasini Teskari matritsa yordamida echish
> with(Student[LinearAlgebra]):
> A:= <<1,2,3,4>|<2,1,2,3>|<3,2,1,2>|<-4,3,2,1>>;
:= A
1 2 3 -4
2 1 2 3
3 2 1 2
4 3 2 1
> B:=<<5,1,1,-5>>;
:= B
5
1
1
-5
> V:=A^(-1);](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_5.png?v=1)
![:= V
-2 1
2 -8 13
2
5
2 -1 21
2 -8
0 1
2 -1 1
2
1
2 0 5
2 -2
> X:=V.B;
:= X
-50
62
-3
15
Tenglamalar sistemasi
Oddiy usul bilan yechish
> solve({3*x+2*y+z=5,2*x+3*y+z=1,2*x+y+3*z=11},[x,y,z]);[ ] [ ] , , x 2 y -2 z 3
Tenglamalar sistemasini Gauss metodi orqali yechish:
> with(LinearAlgebra):
> A:=<<3,2,2>|<2,3,1>|<1,1,3>>;
:= A
3 2 1
2 3 1
2 1 3](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_6.png?v=1)
![> b:=<<5,1,11>>; := b
5
1
11
> GaussianElimination(A);
3 2 1
0 5
3
1
3
0 0 12
5
> GaussianElimination(A,'method'='FractionFree');
3 2 1
0 5 1
0 0 12
> ReducedRowEchelonForm(<A|b>);
1 0 0 2
0 1 0 -2
0 0 1 3
Tenglamalar sistemasini Kramel usulida yechish
> with(Student [LinearAlgebra]):
> A:=<<3,2,2>|<2,3,1>|<1,1,3>>;](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_7.png?v=1)
![Tenglamalarsistemasini Teskari matritsa yordamida echish
> with(Student [LinearAlgebra]):
> A:=<<3,2,2>|<2,3,1>|<1,1,3>>;
:= A
3 2 1
2 3 1
2 1 3
> B:=<<5,1,11>>; := B
5
1
11
> V:=A^(-1);
:= V
2
3 -5
12 -1
12
-1
3 7
12 -1
12
-1
3 1
12 5
12
> X:=V.B;](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_9.png?v=1)
![> with(Student [LinearAlgebra]):
> A:=<<4,1,3>|<-2,1,-2>|<0,2,0>>;
:= A
4 -2 0
1 1 2
3 -2 0
> B:=<<0,2,0>|<-2,4,-3>|<6,3,4>>;
:= B
0 -2 6
2 4 3
0 -3 4
> C:=A^(-1); := C
1 0 -1
3
2 0 -2
-5
4
1
2
3
2
> X:=C.B;
:= X
0 1 2
0 3 1
1 0 0](https://docx.uz/documents/7010bc14-9a30-4067-becb-875fd089ccef/page_10.png?v=1)
Chiziqli tenglamalar sistemasini yechish
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