1.5.43 Systems of Algebraic Equations

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