1.5.37 Systems of algebraic equations
Condition: Given a system of linear equations \( \left\{\begin{array}{c}x_{1}+2 x_{2}+x_{3}+x_{4}=1 \\ x_{1}+3 x_{2}+4 x_{3}=3 \\ 2 x_{1}+5 x_{2}+5 x_{3}+x_{4}=4\end{array}\right. \) a) reduce the extended matrix of the system to the main stepwise form and show that the system is consistent; b) solve the system: write down the general solution; c) find two particular solutions of the system and check one of them.
Solving systems of algebraic equations by the methods of Gauss, Jordan-Gauss, Cramer and using the inverse matrix. Homogeneous and non-homogeneous systems of equations.