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

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: \(\left\{ \begin{array}{l}x_{1}+2x_{2}-x_{3}=4\\3x_{1}+2x_{3}=-8\\4x_{1}-2x_{2}+5x_{3}=0\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x_{1}=4-2x_{2}+x_{3}\\3x_{1}+2x_{3}=-8\\4x_{1}-2x_{2}+5x_{3}=0\end{array}\right.\) - step2: Substitute the value of \(x_{1}:\) \(\left\{ \begin{array}{l}3\left(4-2x_{2}+x_{3}\right)+2x_{3}=-8\\4\left(4-2x_{2}+x_{3}\right)-2x_{2}+5x_{3}=0\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}12-6x_{2}+5x_{3}=-8\\16-10x_{2}+9x_{3}=0\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}x_{2}=\frac{20+5x_{3}}{6}\\16-10x_{2}+9x_{3}=0\end{array}\right.\) - step5: Substitute the value of \(x_{2}:\) \(16-10\times \frac{20+5x_{3}}{6}+9x_{3}=0\) - step6: Multiply the terms: \(16-\frac{5\left(20+5x_{3}\right)}{3}+9x_{3}=0\) - step7: Multiply both sides of the equation by LCD: \(\left(16-\frac{5\left(20+5x_{3}\right)}{3}+9x_{3}\right)\times 3=0\times 3\) - 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 \(x_{3}:\) \(x_{2}=\frac{20+5\times 26}{6}\) - step14: Simplify: \(x_{2}=25\) - step15: Substitute the values of \(x_{2},x_{3}:\) \(x_{1}=4-2\times 25+26\) - step16: Simplify: \(x_{1}=-20\) - step17: Calculate: \(\left\{ \begin{array}{l}x_{1}=-20\\x_{2}=25\\x_{3}=26\end{array}\right.\) - step18: Check the solution: \(\left\{ \begin{array}{l}x_{1}=-20\\x_{2}=25\\x_{3}=26\end{array}\right.\) - step19: Rewrite: \(\left(x_{1},x_{2},x_{3}\right) = \left(-20,25,26\right)\) The solution to the system of equations using the Gauss-Jordan method is \( (x_{1}, x_{2}, x_{3}) = (-20, 25, 26) \).