1.5.40 Systems of algebraic equations
Given a system of linear equations \( \left\{\begin{array}{l}x_{1}+2 x_{2}-10 x_{3}=4 \\ 3 x_{1}+x_{2}-2 x_{3}+x_{4}=3 \\ 4 x_{1}+3 x_{2}-12 x_{3}+x_{4}=7\end{array}\right. \) a) find the rank of the extended matrix of the system and show that the system has infinitely many solutions; b) solve the system: write down the general solution; c) find a particular solution of the system and check it.
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.