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Linear Programming

We have learned that graphing inequalities results in feasible regions representing the different ordered
pairs that satisfy the inequalities.

Graphing inequalities to determine possible solutions to satisfy certain conditions also has real world
applications. Linear programming is a process that often helps businesses make decisions. Linear
programming involves determining inequalities based on certain restrictions or constraints and graphing
those inequalities to determine solutions to satisfy (i.e. the feasible region) and specifically the “best”
solution. The best solution usually being the solution that allows for the most profit or lowest cost.

Ex. # 1 FleetFeet is a shoe company that makes regular and deluxe running shoes. The regular shoes
require 20 minutes of cutting and 30 minutes of sewing. The deluxe shoe requires 40 minutes of cutting
and 20 minutes of sewing. Only 600 minutes of cutting time and 600 minutes of sewing time is available
each day. FleetFeet makes a profit of \$10 on regular shoes and \$12 on deluxe shoes. How many of each
type of shoe should FleetFeet make each day to earn the greatest profit?

To determine the optimal solution the following steps could be followed.

Step 1 - Identify the constraints and express them using inequalities. A constraint is a restriction on the
number of values allowed in the problem.

Let x represent regular shoes                       Let y represent deluxe shoes
1.                                   2.

3.                                  4.

Step 2 – Sketch a graph of the inequalities to determine the feasible region (i.e. the possible ordered
pairs that satisfy all the inequalities). The ordered pair that is the “best” solution always occurs on the
vertices formed by the feasible region so those must be determined. That may require x intercepts and
intersection points to be determined.
Step 3 – Create the equation to be used to determine which ordered pair is the optimal solution. That
equation is often referred to as the Profit Formula since quite often the ordered pair that will provide
the max. profit is what is being sought.

Step 4 Use the equation to determine which ordered pair on the vertices of the feasible region is the
optimal solution (i.e. substitute the ordered pairs into the equation to determine which pair gives the
“best” result).

#2 Matt and Joe do custom paint jobs on motorcycles and cars. Due to the size of their garage, they can
paint a maximum of 8 bikes or a maximum of 4 cars in one day. They can not paint more than 10 in one
day. Matt and Joe earn \$200 for a bike and \$250 for a car. Matt and Joe would like to know how many of
each they should paint in one day to earn the maximum profit.

Let x represent motorcycles                         Let y represent cars

Step 1 –

Step 2 –

Step 3 –

Step 4 -

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