Linear Programming

Linear Programming

Mathematics Grade 10

Objectives

• Use linear programming to find the great and least values to an expression in a region

• Apply linear programming in daily problems

Introduction

• Any linear programming problem will have some fixed restrictions that create inequalities. 

• These can be used to draw a region in the Cartesian plane to reduce the number of possibilities to compare to find the greatest or least value.

Multiple Select

Choose the example about linear programming in daily life (can be more than one)?

• Determining the number of seats for executive and economic class in a train

• Determining the number of parking lots for cars and motorcycle

• Choosing a suitable book with a cheap price

Translating Problems into Mathematical expressions

• The number of students maximum 20: S⩽20

• The number of cars is not less than 100: C⩾100

• Keywords: Maximum, minimum, less than, greater than

Multiple Choice

Which of the following expressions shows the sentence: The number of score (A) maximum is 90.

• A>90

• A<90

• $A\ge0$

• $A\le90$

Multiple Choice

Which of the followings shows the sentence:

The number of cars (C) and motorcycles (M) is not less than 120?

• M+C>120

• M+C<120

• $M+C\le120$

• $M+C\ge120$

Regions in a Plane

• Shading the unwanted region. In this diagram, the inequalities x + 3y ⩽ 4,      x – 4y ⩽ 4 and y – 2x ⩽ 6 

• The possible values for consideration are then left unshaded.

• Hint: You may use geogebra.org  to help you drawing the inequalities

Multiple Choice

Which of these regions would be correct for x + y ⩽ 10 if the unwanted region is shaded?

Multiple Choice

Which of the followings is inside the region of the three inequalities: x + 3y ⩽ 4, x – 4y ⩽ 4 and y – 2x ⩽ 6 ?

•  $\left(5,0\right)$  

•  $\left(0,-2\right)$  

•  $\left(0,10\right)$  

•  $\left(-1,0\right)$  

Greatest and least value

• To find the greatest or least value for an expression it is not necessary to work out every value each time.

• The expression of 2x+y has values shown on the grid.

• When a line is drawn through the zero values it creates a slider that can be used to find the greatest and least values.

Finding the greatest and least value

• The greatest or least values within a particular region can be found by using the line over the region.

• As the line moves across the diagram the last point within the region it touches will be the greatest value.

Multiple Choice

What is the last point that the line 2x+y touches as it moves to the right corner (increasing)?

• (0,4)

• (4,0)

• (-4,-2)

• (-2,2)

Finding the greatest and least values

• In most cases, the last point that the line touches will be one of the corners.

• You can evaluate the expression at each corner point to find which one is the greatest or least value.

Open Ended

Write the vertices (corners)!

Finding the greatest and least values

• Find the greatest and least values of 2x+y

• Check the values of 2x+y using the three corners (4,0), (-2,2) and (-4,-2)

• For (4,0) --> 2x+y --> 2.4+0=8

Multiple Choice

What is the least and the greatest value for

 $2x+y$  using the given corners of the graph?

• The least value = -4, the greatest value = 10

• The least value = 8, the greatest value = 10

• The least value = -4, the greatest value = 10

• The least value = -10, the greatest value = 8

Assessment

• ixl.com, Algebra 2, F.5

• Minimum score = 80

• Due date, September 4th, 2020. At 8 p.m.