Линейни уравнения

[martin_mrv]

−3x1 +2x2 −3x3 +x4 = 1
4x1 −2x2 +2x3 +x4 = 6
3x1 −2x2 −x3 +2x4 = 9
−2x1 +2x2 −x3 +2x4 = 3

Здравейте, може ли да ми помогнете с решаването на тази система, тъй като получавам винаги грешен отговор за нея. Мойте отговори се разминават с 1 единица от долупосочените винаги Благодаря ви много
Ако съм разбрал правилно отговорите са както следва:
х1= 2
х2= 1
х3= -1
х4= 2

[ammornil]

−3x1 +2x2 −3x3 +x4 = 1
4x1 −2x2 +2x3 +x4 = 6
3x1 −2x2 −x3 +2x4 = 9
−2x1 +2x2 −x3 +2x4 = 3
---
[tex]\begin{array} (eq1): \hspace{12} -3.x_1 +2.x_2 -3.x_3 +x_4= 1\\ (eq2): \hspace{12} 4.x_1 -2.x_2 +2.x_3 +x_4= 6 \\ (eq3): \hspace{12} 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \\ (eq4): \hspace{12} -2.x_1 +2.x_2 -x_3 +2.x_4= 3\end{array}[/tex]

Системата се преобразува чрез почленно събиране на уравнения към вида:
[tex]\begin{array} (eq1)+(eq3) \\ (eq2) +2.(eq4) \\ (eq3) -(eq4)\\ (eq3) \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array} -3.x_1 +2.x_2 -3.x_3 +x_4 +3.x_1 -2.x_2 -x_3 +2.x_4= 1 +9 \\ 4.x_1 -2.x_2 +2.x_3 +x_4 +2.(-2.x_1 +2.x_2 -x_3 +2.x_4)= 6 +2.3 \\ 3.x_1 -2.x_2 -x_3 +2.x_4 -(-2.x_1 +2.x_2 -x_3 +2.x_4)= 9 -1 \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array} -4.x_3 +3.x_4 =10 \\ 4.x_1 -2.x_2 +2.x_3 +x_4 -4.x_1 +4.x_2 -2.x_3 +4.x_4= 12 \\ 3.x_1 -2.x_2 -x_3 +2.x_4 +2.x_1 -2.x_2 +x_3 -2.x_4= 8 \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array} -4.x_3 =10 -3.x_4 \\ 2.x_2 +5.x_4 = 12 \\ 5.x_1 -4.x_2 = 8 \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array} 4.x_3 =3.x_4 -10 \\ 2.x_2 = 12 -5.x_4 \\ 5.x_1 = 8 +2.(2.x_2) \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array} 4.x_3 =3.x_4 -10 \\ 2.x_2 = 12 -5.x_4 \\ 5.x_1 = 8 +2.(12 -5.x_4) \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array} x_3 =\frac{3.x_4-10}{4} \\ x_2 = \frac{12 -5.x_4}{2} \\ x_1 =\frac{\cancel{5}.(6-2.x_4)}{\cancel{5}} \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ 3.x_1 -2.x_2 -x_3 +2.x_4= 9 \end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ 3.(6-2.x_4) -\cancel{2}.\frac{12 -5.x_4}{\cancel{2}} -\frac{3.x_4-10}{4} +2.x_4= 9 \end{array}\hspace{5} \Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ 18 -6.x_4 -12 +5.x_4+2.x_4 -\frac{3.x_4-10}{4} -9=0\end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ x_4 -3 -\frac{3.x_4-10}{4} =0 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ 4.x_4 -12 -3.x_4 +10 =0 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array}x_1= 6-2.x_4 \\ x_2 = \frac{12 -5.x_4}{2} \\ x_3 =\frac{3.x_4-10}{4} \\ x_4 =2 \end{array}[/tex]

[tex]\Rightarrow \hspace{5} \begin{array}x_1= 6-2.2 \\ x_2 = \frac{12 -5.2}{2} \\ x_3 =\frac{3.2-10}{4} \\ x_4 =2 \end{array} \hspace{5} \Rightarrow \hspace{5} \begin{array}x_1= 2 \\ x_2 = 1 \\ x_3 =-1 \\ x_4 =2 \end{array}[/tex]