1.5.43 Systems of algebraic equations

Condition: Given a system of linear equations: \( \left\{\begin{array}{l}2 x_{1}-3 x_{2}+x_{3}-6 x_{4}=-3 \\ x_{1}+x_{2}+x_{3}-2 x_{4}=0 \\ 3 x_{1}-2 x_{2}+2 x_{3}-8 x_{4}=-3\end{array}\right. \) a) show that the system has infinitely many solutions; b) solve the system: write down the general solution, find a non-trivial particular solution and check it; c) write down the general solution of the corresponding homogeneous system \[ \left\{\begin{array}{l} 2 x_{1}-3 x_{2}+x_{3}-6 x_{4}=0 \\ x_{1}+x_{2}+x_{3}-2 x_{4}=0 \\ 3 x_{1}-2 x_{2}+2 x_{3}-8 x_{4}=0 \end{array}\right. \]