The graphical solution of linear programming problems is based on the geometric analysis of inequalities and the objective function in systems of two variables (n=2). Each inequality represents a half-plane bounded by a straight line, and the conditions represent areas bounded by the coordinate axes. The solution domain of the system, which can have a common solution, is formed as a convex polygon, a single point, or an empty set. The graphical method includes the following steps: constructing graphs of inequalities, determining the solution domain, drawing the objective function vector, finding the extreme point by parallel displacement in the direction of the vector, and calculating the optimal solution. In the example, the system of inequalities is analyzed graphically, and the maximum (3,0) and minimum (0,3) values of the Z function are determined. This method allows you to solve the problem visually and intelligibly, and practically demonstrates the main principles of linear programming.

Linear programming, graphical method, system of inequalities, domain of possible solutions, convex polygon, objective function, extremum point, vector direction, optimal solution, half-plane, coordinate axes, maximum value, minimum value.

Yemelichev, VA, Kovalev, MM, Kravtsov, MK (1984). *Mathematicheskie metody programming*. Moscow: Nauka.

Bertsimas, D., Tsitsiklis, JN (1997). *Introduction to Linear Optimization*. Belmont, MA: Athena Scientific.

Taha, HA (2017). *Operations Research: An Introduction* (10th ed.). Pearson.

Kholmirzayev, AA, Kuldoshev, Q. (2010). *Fundamentals of Mathematical Programming*. Tashkent: Publishing House of the National University of Uzbekistan.

Anderson, DR, Sweeney, DJ, Williams, TA (2018). *An Introduction to Management Science: Quantitative Approaches to Decision Making* (15th ed.). Cengage Learning.