In linear programming, the feasible region is a convex set because the constraints are all linear functions. Referring to the graphic method of two-dimensional problem, its feasible region is a region surrounded by several lines, so it must be a convex set. Then, the optimal solution is searched in this convex set. If the solution is searched by moving the isoline of the objective function, then the optimal solution must reach the optimal value at the edge of its convex set, and the edge of the convex set is either a line segment or a vertex, so the optimal solution of the linear programming problem must be at the vertex of the feasible region.

In fact, these vertices are the basic feasible solutions of linear programming problems.

So how do you get these vertices (basically feasible solutions) from the model?

The key to solving the model is to solve ax = b.

Because a matrix is an m×n matrix, the unique solution of the above constraint equation cannot be obtained. The nonsingular submatrix b of m×m must be found in the A matrix, that is, it satisfies that | b | is not equal to zero (the determinant is not zero), so that the unique solution of bx = b can be obtained. At this time, the decision variables corresponding to matrix B are called base variables, and the rest are non-base variables. In x, if the basic variable takes the value of BX = B and the non-basic variable takes the value of zero, then x is the basic (feasible) solution of the problem, that is, the solution corresponding to the vertex of the row field.

This is written according to my understanding, I hope it helps.