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**Solve problems by using linear programming.**

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Linear Programming Method for finding a minimum or maximum value given constraints(limits). The constraints come from a system of linear inequalities. The graph of the system is call the feasible region.

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Objective Function Models the quantity you are trying to maximize or minimize. Often cost or profit Ex: the objective function is C = 2x + y Graphs of lines with various values for C are parallel If there is a max or min for the objective function, it occurs at one or more of the vertices of the feasible region. This is called the Vertex Principle of Linear Programming

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Testing Vertices What point in the feasible region maximizes P for the objective function P = 2x + y? Graph the system to find the feasible region. Vertices: (0,2.5) (3,1) (2,0) (0,0) Evaluate P at each vertex. P has a maximum value of 7 at (3,1).

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**Using Linear Programming**

You have at most 20 hours to make T-shirts &sweatshirts. You want to spend no more than $600 on supplies. You want to have at least 50 items to sell.

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**Continued How many of each to maximize profit? Create the constraints:**

So the objective function is: P = 6x + 20y Graph constraints: Test vertices: (50,0) (25,25) (75,15) (120,0) P is maximized at (75,15) Sell 75 t-shirts and 15 sweatshirts.

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Assignment Odds p.160 #11-23

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