Chapter 3

FInd corner points using z = 4x + 2y

(0,8) | 16 (3,4) | 20 (13/2,2) | 30 (12,0) | 48

Graph feasible region x + 3y ≤ 6 2x + 4y ≥ 7

7/4 , 7/2

Corner Point Theorem

If an optimum value of the objective function exists, it will occur at one of the corner points of the feasible region

Linear Inequality

This is a linear function that consists of the greater than less than or equal to symbols.

Constraints

a set of restrictions written as equasions

Feasible Region

area where all the inequalities work

Find max and min z = 3x + 2y z = x + 4y

max 29 @ (7,4) min 10 @ (0,5)

using corner points (0,8) (3,4) (13/2,2) (12,0) and z = 2x + 3y find the max and min

max = none min = 18 @ (3,4)

maximize z = 2x + 4y subject to 3x + 2y ≤ 12 5x + y ≥ 5 x ≥ 0 y ≥ 0

max of 24 @ (0,6)

Objective Function

maximum or minimum value of a function

using corner points (6,0) and (3,4) with z= 4x +7y find the max and min

min 24 @ (6,0) max 40 @ (3,4)

using corner points (0,0) (8,0) (7,3) (4,8) (0,12) and z = 1.5x + .25y find the max and min

min = 0 @ (0,0) max = 12 @ (8,0)

Minimize z = 4x + 7y subject to: x - y ≥ 1 3x +2y ≥ 18 x ≥ 0 y ≥ 0

minimum of 24 when x = 6 and y = 0

Half - Plane

region on one side of the line

graph linear inequality x + y ≤ 2

rise over run = -2/2

Boundaries

separates the points in the solution from the points that are not a solution

Corner Point

this is where the boundary lines of two constraints cross in the feasible region

System of Inequalities

two or more inequalities

Unbounded

when the feasible region has a side(s) that is not closed off.

Bounded

when the feasible region is closed off by boundaries on all sides

what do variables stand for Y = mx + b

y= y variable m= slope b= y intercept x= x variable