What does indecisiveness lead to?

variables, and the corresponding definitions are transferred by analogy to the general case.

On the other hand, the value of a geometric language lies in the fact that intuition can suggest some constructions and methods for solving problems that are difficult to figure out if you use only algebraic language. The Simpleskmethod provides a good example of such a technique. Because of this, it turns out to be useful to introduce geometric language in the case of an arbitrary number of variables.

Exercises

1. Prove that if the planes
   $$x_1 + x_2 + x_3 = K$$
   are parallel to some face of the polyhedron $A$, then the plane
   $$x_1 + x_2 + x_3 = K_0$$
   coincides with the plane of this face.

2. Is it true that if the planes
   $$x_1 + x_2 + x_3 = K$$
   are parallel to some edge of the polyhedron $A$, then the plane
   $$x_1 + x_2 + x_3 = K_0$$
   goes through this rib?

3. a) Let the problem of choosing technological resources set the task of maximizing not the output, but the approximation, that is, the difference between the price of the produced product and the cost of using the resource in production, and it is known that the price of a unit of produced product is equal to $q$ rubles, the cost of using units of the $j$ resource is equal to $q_j$ rubles ($j=1,\dots,m$).

6. Let the choice problem be technological, so that the output is no less than $S$ units and at the same time that the costs are minimal.

Formulate each of these problems as a linear programming problem. Prove that there is exactly one value $S$ for which the solutions to problem 6) coincide with the solutions to problem a).

Literature

A. S. Solodovnikov, Introduction to linear algebra and linear programming. M., “Enlightenment”, 1966.