If linear algebra grew out of the solution of systems of linear equations, then linear programming grew out of attempts to solve systems of linear inequalities, allowing one to optimise linear functions subject to constraints expressed as inequalities. …

The linear programming problem can be solved using different methods, such as the graphical method, simplex method, or by using tools such as R, open solver etc. Here, we will discuss the two most important techniques called the simplex method and graphical method in detail.

Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics).

Linear programming is a way of using systems of linear inequalities to find a maximum or minimum value. In geometry, linear programming analyzes the vertices of a polygon in the Cartesian plane. Linear programming is one specific type of mathematical optimization, which has applications in many scientific fields.

The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables. The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.

There are two methods that will be used in this lesson to solve a system of linear equations algebraically. They are 1) substitution, and 2) elimination. They are both aimed at eliminating one variable so that normal algebraic means can be used to solve for the other variable.

Linear programming is used for obtaining the most optimal solution for a problem with given constraints. In linear programming, we formulate our real-life problem into a mathematical model. It involves an objective function, linear inequalities with subject to constraints.

One technique is the simplex method, which was developed in the late 1940s by George Dantzig and is based on the Gauss–Jordan elimination method. The simplex method is readily adaptable to the computer, which makes it suitable for solving linear programming problems involving large numbers of variables and constraints.

A linear programming problem will have no solution if the simplex method breaks down at some stage. For example, if at some stage there are no nonnegative ratios in our computation, then the linear programming problem has no solution. 4.2 The Simplex Method: Standard Minimization Problems

Given our incapacity to plot graphs beyond two (nowadays three) dimensions, we resort to algebra to find out if we can may be find out the solution without the need to actually plot the graph.

The first and the second columns list the basic and the non basic variables respectively. The third column lists the solution of the basic variables in the same order as they are mentioned in the first column and the fourth column lists the Z value for the corresponding solution.