hiziqli tenglamalar sistemasini Gauss usulida yeching $$ \left\{\begin{array}{l}x_{1}+2 x_{2}-x_{3}=4 \ 3 x_{1}+2 x_{3}=-8 \ 4 x_{1}-2 x_{2}+5 x_{3}=0\end{array}\right. $$

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Solve the system of equations $$ x_{1}+2x_{2}-x_{3}=4;3x_{1}+2x_{3}=-8;4x_{1}-2x_{2}+5x_{3}=0 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x_{1}+2x_{2}-x_{3}=4\3x_{1}+2x_{3}=-8\4x_{1}-2x_{2}+5x_{3}=0\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x_{1}=4-2x_{2}+x_{3}\3x_{1}+2x_{3}=-8\4x_{1}-2x_{2}+5x_{3}=0\end{array}ight.$$
- step2: Substitute the value of $$x_{1}:$$
  $$\left\{ \begin{array}{l}3\left(4-2x_{2}+x_{3}ight)+2x_{3}=-8\4\left(4-2x_{2}+x_{3}ight)-2x_{2}+5x_{3}=0\end{array}ight.$$
- step3: Simplify:
  $$\left\{ \begin{array}{l}12-6x_{2}+5x_{3}=-8\16-10x_{2}+9x_{3}=0\end{array}ight.$$
- step4: Solve the equation:
  $$\left\{ \begin{array}{l}x_{2}=\frac{20+5x_{3}}{6}\16-10x_{2}+9x_{3}=0\end{array}ight.$$
- step5: Substitute the value of $$x_{2}:$$
  $$16-10	imes \frac{20+5x_{3}}{6}+9x_{3}=0$$
- step6: Multiply the terms:
  $$16-\frac{5\left(20+5x_{3}ight)}{3}+9x_{3}=0$$
- step7: Multiply both sides of the equation by LCD:
  $$\left(16-\frac{5\left(20+5x_{3}ight)}{3}+9x_{3}ight)	imes 3=0	imes 3$$
- step8: Simplify the equation:
  $$-52+2x_{3}=0$$
- step9: Move the constant to the right side:
  $$2x_{3}=0+52$$
- step10: Remove 0:
  $$2x_{3}=52$$
- step11: Divide both sides:
  $$\frac{2x_{3}}{2}=\frac{52}{2}$$
- step12: Divide the numbers:
  $$x_{3}=26$$
- step13: Substitute the value of $$x_{3}:$$
  $$x_{2}=\frac{20+5	imes 26}{6}$$
- step14: Simplify:
  $$x_{2}=25$$
- step15: Substitute the values of $$x_{2},x_{3}:$$
  $$x_{1}=4-2	imes 25+26$$
- step16: Simplify:
  $$x_{1}=-20$$
- step17: Calculate:
  $$\left\{ \begin{array}{l}x_{1}=-20\x_{2}=25\x_{3}=26\end{array}ight.$$
- step18: Check the solution:
  $$\left\{ \begin{array}{l}x_{1}=-20\x_{2}=25\x_{3}=26\end{array}ight.$$
- step19: Rewrite:
  $$\left(x_{1},x_{2},x_{3}ight) = \left(-20,25,26ight)$$
The solution to the system of equations using the Gauss-Jordan method is $$ (x_{1}, x_{2}, x_{3}) = (-20, 25, 26) $$.
The solution is $$ x_{1} = -20 $$, $$ x_{2} = 25 $$, and $$ x_{3} = 26 $$.

hiziqli tenglamalar sistemasini Gauss usulida yeching

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