Q:

Switching the solutions in linear programming.

Reduce the feasible solution $x_1=2,x_2=1,x_3=1$ for the linear programming problem $$
\begin{split}
\max\      & x_1+2x_2+3x_3\\
\text{subject to }\ &  x_1 - x_2 + 3x_3 &= 4\\
                    & 2x_1 + x_2 +  x_3 &= 6\\
                    & x_1,x_2,x_3 &\ge 0
\end{split}
$$

According to definition of Basic feasible solution, i think the given solution is already basic, please clarify someone..the two concepts.

A:

Geometrically, a basic solution would be a corner point of the underlying polyhedron. Algebraically, since you are in $\mathbb{R}^3$, a basic solution would satisfy three constraints with equality.
You have 5 constraints, can you check how many of them are satisfied with equality?

