1.5.19 Systems of algebraic equations
Condition: Reduce the system of equations to an equivalent allowed system by including \( x_{1}, x_{2}, x_{3} \) in the set of allowed unknowns. Write down the general solution, find the corresponding basic solution. Resolve the system and write a new general and corresponding basis solution. \[ \left\{\begin{array}{c} 2 x_{1}+3 x_{2}+4 x_{3}+9 x_{4}+16 x_{5}=1 \\ x_{1}+2 x_{2}+2 x_{3}+5 x_{4}+9 x_{5}=1 \\ 3 x_{1}+3 x_{2}+5 x_{3}+11 x_{4}+20 x_{5}=2 \end{array}\right. \]
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.