30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Problem Solving with
Linear Programming
LP-L5 Objectives:
To solve complex problems using Linear
Programming techniques.
Learning Outcome B-1
Slide 1
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
The process of finding a feasible region and locating the points
that give the minimum or maximum value to a specific
expression is called linear programming. It is frequently used
to determine maximum profits, minimum costs, minimum
distances, and so on.
Theory – Intro
Slide 2
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values
of x and y that maximize the expression M = x + 3y.
Example - Maximize the Value of
a Specific Expression
x+y6
x + 2y  8
x2
y1
Slide 3
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values of
x and y that maximize the expression M = x + 3y.
Solution
1. Graph the system: The feasible region is the green shaded
area shown
2. Find the vertices of the feasible
region: The coordinates of the corner
points are (2, 3), (2, 1), (5, 1), and (4, 2).
3. Substitute each vertice into the
equation to find maximum:
The value of M for each point is
Point (2, 3): M = 2 + 3(3) = 11
Point (2, 1): M = 2 + 3(1) = 5
Point (5, 1): M = 5 + 3(1) = 8
Point (4, 2): M = 4 + 3(2) = 10
Therefore, the value of M is maximized at (2, 3).
Example - Maximize the Value of
a Specific Expression
x+y6
x + 2y  8
x2
y1
Slide 4
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values
of x and y that maximize the expression M = 4x + y.
Test Yourself - Maximize the
Value of a Specific Expression
x0
y0
3x + 2y  6
2x + 3y  6
Slide 5
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values
of x and y that maximize the expression M = 4x + y.
Solution
1. Graph the system: The feasible region is the green shaded
area shown
2. Find the vertices of the feasible
region: The coordinates of the corner
points are (0, 3), (0, 2), and (1.2, 1.2).
3. Substitute each vertice into the
equation to find maximum:
Using (0, 3), M = 4(0) + 3 = 3.
Using (0, 2), M = 4(0) + 2 = 2.
Using (1.2, 1.2), M = 4(1.2) + 1.2 = 6.
x0
y0
3x + 2y  6
2x + 3y  6
The coordinates (1.2, 1.2) produce the maximum value of the
expression 4x + y.
Test Yourself - Maximize the
Value of a Specific Expression
Slide 6
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values
of x and y that minimize the expression M = 3x + 2y.
Test Yourself – Minimize the
Value of a Specific Expression
x+y4
x + 5y  8
-x + 2y  6
Slide 7
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Graph the following system of inequalities and identify the
corner points of the feasible region. Then find the values
of x and y that minimize the expression M = 3x + 2y.
x+y4
x + 5y  8
-x + 2y  6
Solution
1. Graph the system: The feasible region is the green shaded
area shown
2. Find the vertices of the feasible
region: The coordinates of the corner
points are (-2, 2), (3, 1), and (0.67, 3.33).
3. Substitute each vertice into the
equation to find minimum:
Using (-2, 2), M = 3(-2) + 2(2) = -2.
Using (3, 1), M = 3(3) + 2(1) = 11.
Using (0.67, 3.33), M = 3(0.67) + 2(3.33) = 8.67.
The coordinates (-2, 2) produce the minimum value of the
expression 3x + 2y.
Test Yourself – Minimize the
Value of a Specific Expression
Slide 8
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
The constraints for manufacturing two types of hockey skates
are given by the following system of inequalities. Find the
maximum value of Q over the feasible region if Q = 3x + 5y.
Test Yourself – Maximize the
Value of a Specific Expression
y  -1x + 4
x + 4y  7
-x + 2y  5
Slide 9
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
The constraints for manufacturing two types of hockey skates
are given by the following system of inequalities. Find the
maximum value of Q over the feasible region if Q = 3x + 5y.
y  -1x + 4
x + 4y  7
-x + 2y  5
Solution
1. Graph the system: The feasible region is the green shaded
area shown
2. Find the vertices of the feasible
region: The coordinates of the corner
points are (1, 3), (-1, 2), and (3, 1).
3. Substitute each vertice into the
equation to find maximum:
Using (1, 3), Q = 3(1) + 5(3) = 18.
Using (-1, 2), Q = 3(-1) + 5(2) = 7.
Using (3, 1), Q = 3(3) + 5(1) = 14.
The coordinates (1, 3) produce a maximum value for Q over
the feasible region where Q = 3x + 5y.
Test Yourself – Maximize the
Value of a Specific Expression
Slide 10
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 5: Problem Solving
Here is a plan of the steps used to solve word problems
using linear programming:
1. After reading the question, make a chart to see the
information more clearly.
2. Assign variables to the unknowns.
3. Form expressions to represent the restrictions.
4. Graph the inequalities.
5. Find the coordinates of the corner points of the feasible
region.
6. Find the vertex point that maximizes or minimizes what we
are looking for.
7. State the solution in a sentence.
Theory – Solving Problems
Using Linear Programming
Slide 11
30S Applied Math
Mr. Knight – Killarney School
Example – Seven Steps
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 12
30S Applied Math
Mr. Knight – Killarney School
Example – Seven Steps cont’d
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 13
30S Applied Math
Mr. Knight – Killarney School
Example – Seven Steps cont’d
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 14
30S Applied Math
Mr. Knight – Killarney School
Example 2 – Seven Steps
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 15
30S Applied Math
Mr. Knight – Killarney School
Example 2 – Seven Steps cont’d
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 16
30S Applied Math
Mr. Knight – Killarney School
Example 2 – Seven Steps cont’d
Unit: Linear Programming
Lesson 5: Problem Solving
Slide 17