N-R(A)= 1, which means there is only one free variables. In addition, if you look at the last line, it is equivalent to equation 7X4=0. X4 obviously cannot be assigned any other value.

It is not acceptable to assign all free variables to 0. Free variables should take a set of linearly independent vectors such as (1, 0,0), (0, 1, 0), (0,0, 1), and then get the basic solution system from the same solution system. The group of linearly independent vectors is still linearly independent after adding some components, so free variables takes a group of linearly independent vectors and adds the value of the upper constraint variable.

free variables

(free variables) refers to the design variables without nonnegative conditions in linear programming. If the problem contains such variables, in order to form the standard formula of linear programming, two subtracted non-negative design variables are often used instead, so that all design variables in the mathematical model of optimal design are non-negative design variables. The optimal solutions before and after replacement are equivalent. Linear programming problems with free variables can sometimes be transformed into dual problems by dual theory and solved directly.