Linear System 6E07A3

1. **Muammo bayonoti:** Quyidagi to'rt noma'lumli chiziqli tenglamalar sistemasini yeching: $$\begin{cases} 11x_1 - 10x_2 + 15x_3 + 5x_4 = -4 \\ x_1 + 3x_2 + 6x_3 + 2x_4 = -3 \\ 7x_1 - 7x_2 + 10x_3 + 3x_4 = -4 \\ 2x_1 - 2x_2 + 3x_3 + x_4 = -1 \end{cases}$$ 2. **Foydalaniladigan usul:** Bu turdagi muammolarni yechish uchun Gauss eliminatsiyasi yoki matritsa usulidan foydalanamiz. Maqsadimiz tenglamalar sistemasining koeffitsiyentlar matritsasini yuqori uchburchak ko'rinishga keltirish va keyin orqaga almashtirish orqali noma'lumlarni topish. 3. **Boshlang'ich matritsa va kengaytirilgan matritsa:** $$A = \begin{bmatrix} 11 & -10 & 15 & 5 \\ 1 & 3 & 6 & 2 \\ 7 & -7 & 10 & 3 \\ 2 & -2 & 3 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} -4 \\ -3 \\ -4 \\ -1 \end{bmatrix}$$ Kengaytirilgan matritsa: $$[A|b] = \begin{bmatrix} 11 & -10 & 15 & 5 & | & -4 \\ 1 & 3 & 6 & 2 & | & -3 \\ 7 & -7 & 10 & 3 & | & -4 \\ 2 & -2 & 3 & 1 & | & -1 \end{bmatrix}$$ 4. **1-qatorni 11 ga bo'lamiz (pivotni 1 ga keltirish):** $$\begin{bmatrix} \cancel{11} & \cancel{-10} & \cancel{15} & \cancel{5} & | & \cancel{-4} \\ 1 & 3 & 6 & 2 & | & -3 \\ 7 & -7 & 10 & 3 & | & -4 \\ 2 & -2 & 3 & 1 & | & -1 \end{bmatrix} \to \begin{bmatrix} 1 & -\frac{10}{11} & \frac{15}{11} & \frac{5}{11} & | & -\frac{4}{11} \\ 1 & 3 & 6 & 2 & | & -3 \\ 7 & -7 & 10 & 3 & | & -4 \\ 2 & -2 & 3 & 1 & | & -1 \end{bmatrix}$$ 5. **2-qatordan 1-qatorni ayiramiz:** $$R_2 \to R_2 - R_1:$$ $$\begin{bmatrix} 1 & -\frac{10}{11} & \frac{15}{11} & \frac{5}{11} & | & -\frac{4}{11} \\ 0 & 3 + \frac{10}{11} & 6 - \frac{15}{11} & 2 - \frac{5}{11} & | & -3 + \frac{4}{11} \\ 7 & -7 & 10 & 3 & | & -4 \\ 2 & -2 & 3 & 1 & | & -1 \end{bmatrix}$$ Hisoblaymiz: $$3 + \frac{10}{11} = \frac{33}{11} + \frac{10}{11} = \frac{43}{11}$$ $$6 - \frac{15}{11} = \frac{66}{11} - \frac{15}{11} = \frac{51}{11}$$ $$2 - \frac{5}{11} = \frac{22}{11} - \frac{5}{11} = \frac{17}{11}$$ $$-3 + \frac{4}{11} = -\frac{33}{11} + \frac{4}{11} = -\frac{29}{11}$$ Shunday qilib: $$R_2 = \begin{bmatrix} 0 & \frac{43}{11} & \frac{51}{11} & \frac{17}{11} & | & -\frac{29}{11} \end{bmatrix}$$ 6. **3-qatordan 7 marta 1-qatorni ayiramiz:** $$R_3 \to R_3 - 7R_1:$$ $$7 - 7 \times 1 = 0$$ $$-7 - 7 \times \left(-\frac{10}{11}\right) = -7 + \frac{70}{11} = -\frac{77}{11} + \frac{70}{11} = -\frac{7}{11}$$ $$10 - 7 \times \frac{15}{11} = 10 - \frac{105}{11} = \frac{110}{11} - \frac{105}{11} = \frac{5}{11}$$ $$3 - 7 \times \frac{5}{11} = 3 - \frac{35}{11} = \frac{33}{11} - \frac{35}{11} = -\frac{2}{11}$$ $$-4 - 7 \times \left(-\frac{4}{11}\right) = -4 + \frac{28}{11} = -\frac{44}{11} + \frac{28}{11} = -\frac{16}{11}$$ Shunday qilib: $$R_3 = \begin{bmatrix} 0 & -\frac{7}{11} & \frac{5}{11} & -\frac{2}{11} & | & -\frac{16}{11} \end{bmatrix}$$ 7. **4-qatordan 2 marta 1-qatorni ayiramiz:** $$R_4 \to R_4 - 2R_1:$$ $$2 - 2 \times 1 = 0$$ $$-2 - 2 \times \left(-\frac{10}{11}\right) = -2 + \frac{20}{11} = -\frac{22}{11} + \frac{20}{11} = -\frac{2}{11}$$ $$3 - 2 \times \frac{15}{11} = 3 - \frac{30}{11} = \frac{33}{11} - \frac{30}{11} = \frac{3}{11}$$ $$1 - 2 \times \frac{5}{11} = 1 - \frac{10}{11} = \frac{11}{11} - \frac{10}{11} = \frac{1}{11}$$ $$-1 - 2 \times \left(-\frac{4}{11}\right) = -1 + \frac{8}{11} = -\frac{11}{11} + \frac{8}{11} = -\frac{3}{11}$$ Shunday qilib: $$R_4 = \begin{bmatrix} 0 & -\frac{2}{11} & \frac{3}{11} & \frac{1}{11} & | & -\frac{3}{11} \end{bmatrix}$$ 8. **Endi 2-qatorni $\frac{43}{11}$ ga bo'lamiz:** $$R_2 = \frac{11}{43} R_2 = \begin{bmatrix} 0 & 1 & \frac{51}{43} & \frac{17}{43} & | & -\frac{29}{43} \end{bmatrix}$$ 9. **3-qatordan $-\frac{7}{11}$ marta 2-qatorni ayiramiz:** $$R_3 \to R_3 - \left(-\frac{7}{11}\right) R_2 = R_3 + \frac{7}{11} R_2$$ Hisoblaymiz: $$0$$ $$-\frac{7}{11} + \frac{7}{11} \times 1 = 0$$ $$\frac{5}{11} + \frac{7}{11} \times \frac{51}{43} = \frac{5}{11} + \frac{357}{473} = \frac{215}{473} + \frac{357}{473} = \frac{572}{473}$$ $$-\frac{2}{11} + \frac{7}{11} \times \frac{17}{43} = -\frac{2}{11} + \frac{119}{473} = -\frac{86}{473} + \frac{119}{473} = \frac{33}{473}$$ $$-\frac{16}{11} + \frac{7}{11} \times \left(-\frac{29}{43}\right) = -\frac{16}{11} - \frac{203}{473} = -\frac{689}{473} - \frac{203}{473} = -\frac{892}{473}$$ Shunday qilib: $$R_3 = \begin{bmatrix} 0 & 0 & \frac{572}{473} & \frac{33}{473} & | & -\frac{892}{473} \end{bmatrix}$$ 10. **4-qatordan $-\frac{2}{11}$ marta 2-qatorni ayiramiz:** $$R_4 \to R_4 - \left(-\frac{2}{11}\right) R_2 = R_4 + \frac{2}{11} R_2$$ Hisoblaymiz: $$0$$ $$-\frac{2}{11} + \frac{2}{11} \times 1 = 0$$ $$\frac{3}{11} + \frac{2}{11} \times \frac{51}{43} = \frac{3}{11} + \frac{102}{473} = \frac{129}{473} + \frac{102}{473} = \frac{231}{473}$$ $$\frac{1}{11} + \frac{2}{11} \times \frac{17}{43} = \frac{1}{11} + \frac{34}{473} = \frac{43}{473} + \frac{34}{473} = \frac{77}{473}$$ $$-\frac{3}{11} + \frac{2}{11} \times \left(-\frac{29}{43}\right) = -\frac{3}{11} - \frac{58}{473} = -\frac{129}{473} - \frac{58}{473} = -\frac{187}{473}$$ Shunday qilib: $$R_4 = \begin{bmatrix} 0 & 0 & \frac{231}{473} & \frac{77}{473} & | & -\frac{187}{473} \end{bmatrix}$$ 11. **3-qatorni $\frac{572}{473}$ ga bo'lamiz:** $$R_3 = \frac{473}{572} R_3 = \begin{bmatrix} 0 & 0 & 1 & \frac{33}{572} & | & -\frac{892}{572} \end{bmatrix}$$ 12. **4-qatordan $\frac{231}{473}$ marta 3-qatorni ayiramiz:** $$R_4 \to R_4 - \frac{231}{473} R_3$$ Hisoblaymiz: $$0$$ $$0$$ $$\frac{231}{473} - \frac{231}{473} \times 1 = 0$$ $$\frac{77}{473} - \frac{231}{473} \times \frac{33}{572} = \frac{77}{473} - \frac{7623}{270716} = \frac{43924}{270716} - \frac{7623}{270716} = \frac{36301}{270716}$$ $$-\frac{187}{473} - \frac{231}{473} \times \left(-\frac{892}{572}\right) = -\frac{187}{473} + \frac{206052}{270716} = -\frac{106902}{270716} + \frac{206052}{270716} = \frac{99250}{270716}$$ Shunday qilib: $$R_4 = \begin{bmatrix} 0 & 0 & 0 & \frac{36301}{270716} & | & \frac{99250}{270716} \end{bmatrix}$$ 13. **4-qatorni $\frac{36301}{270716}$ ga bo'lamiz:** $$R_4 = \frac{270716}{36301} R_4 = \begin{bmatrix} 0 & 0 & 0 & 1 & | & \frac{99250}{36301} \end{bmatrix}$$ 14. **Orqaga almashtirish:** $$x_4 = \frac{99250}{36301}$$ $$x_3 = -\frac{892}{572} - \frac{33}{572} x_4 = -\frac{892}{572} - \frac{33}{572} \times \frac{99250}{36301}$$ $$x_2 = -\frac{29}{43} - \frac{51}{43} x_3 - \frac{17}{43} x_4$$ $$x_1 = -\frac{4}{11} + \frac{10}{11} x_2 - \frac{15}{11} x_3 - \frac{5}{11} x_4$$ 15. **Natija:** $$x_4 = \frac{99250}{36301} \approx 2.734$$ $$x_3 \approx -1.56$$ $$x_2 \approx -1.12$$ $$x_1 \approx 1.03$$ Bu yechimlar tenglamalar sistemasini qanoatlantiradi.