# Linear algebra

Jul 28th, 2016
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notes on linear programming under the subject linear algebra

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Linear Programming:Linear programming is the process of taking various linear inequalities relating to some situation, andfinding the "best" value obtainable under those conditions. A typical example would be taking thelimitations of materials and labor, and then determining the "best" production levels for maximal profitsunder those conditions.In "real life", linear programming is part of a very important area of mathematics called "optimizationtechniques". This field of study (or at least the applied results of it) are used every day in the organizationand allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more.In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.The general process for solving linear-programming exercises is to graph the inequalities (called the"constraints") to form a walled-off area on the x,y-plane (called the "feasibility region"). Then you figureout the coordinates of the corners of this feasibility region (that is, you find the intersection points of thevarious pairs of lines), and test these corner points in the formula (called the "optimization equation") forwhich you're trying to find the highest or lowestvalue.Find the maximal and minimal value ofz = 3x + 4y subject to the followingconstraints:The three inequalities in the curly bracesare the constraints. The area of the planethat they mark off will be the feasibilityregion. The formu

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