1.5.52 Systems of algebraic equations
Condition: Explore the system of equations for consistency and certainty without solving it. Indicate the main (basic) and free variables. \[ \left\{\begin{array}{c} x_{1}+2 x_{2}-3 x_{3}-2 x_{4}=5 \\ -2 x_{1}+x_{3}+4 x_{4}=0 \\ -3 x_{1}+2 x_{2}+4 x_{3}+3 x_{4}=-11 \\ 13 x_{1}-7 x_{3}-9 x_{4}=35 \end{array}\right. \]
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.