Algebra 2: Chapter 3
3.5: Linear Programming
Warm Up
1.
x y 7
Solve: 
2 x  y  8
Objectives
 Write and graph a set of constraints for a linear-programming problem.
 Use linear programming to find the maximum and minimum value of an objective function.
Linear Programming:
Constraints:
Feasible Region:
Objective Function:
Corner-Point Principle:
1.
a.
Graph the feasible region for the
 y  2x  2
 yx5

set of constraints: 
.
 x0
 y  1
b.
Find the vertices of the feasible region.
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2.
The feasible region for a set of constraints has vertices (3, 0), (4, 3), (-1, 6) and (-4, 0).
Given this feasible region, find the maximum and minimum values of each objective function.
a.
H = 3x + y
b.
I = -2x + 3y
Maximum:
Maximum:
Minimum:
Minimum:
3.
Find the maximum and minimum values, if they exist, of each objective function for the
given constraints.
a.
R = 2x – y
b.
M = -3x + y
Constraints:
Constraints:
y  x  1
 y  1x  1


 y0
 x  4
x  y  5
x  y  6


 x3
 x  5
Maximum:
Maximum:
Minimum:
Minimum:
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4.
A small company produces knitted afghans and sweaters and sells them through a chain of
specialty stores. The company is to supply the stores with a total of no more than 100 afghans
and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60
afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each
afghan and a profit of $12 on each sweater.
a.
Define the variables to be used for this situation.
b.
Write a system of inequalities to represent the constraints.
c.
Graph the feasible region.
d.
Find the vertices of the feasible region.
e.
Write an objective function for the company’s total profit, P, from the sale of
afghans and sweaters.
f.
Find the maximum profit that the company can make in a day.
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