Lecture 7.1 to 7.2 bt

Today’s Agenda
 Attendance / Announcements
 Questions from 6.1 / 6.2
 Sections 7.1 / 7.2
 E.C. Quiz Today

Exam Schedule
 Exam 5 (Ch 6.1, 7)
Wed 4/1
 Exam 6 (Ch 10)
Monday 4/27
 Final Exam (Cumulative)
Monday 5/4

Graphing Systems of Inequalities
1. Graph boundary lines
2. Check inequality signs
(Dashed or Solid?)
3. Shade accordingly
5. Test Points
4𝑥 + 𝑦 ≥ 9
2𝑥 + 3𝑦 ≤ 7
4. Identify Intersecting
regions (“Feasible”)

Graph the Feasible Region
5𝑦 − 2𝑥 ≤ 10
𝑥 ≥ 3
𝑦 ≥ 2
𝑦 ≤
2
5
𝑥 + 2
𝑥 ≥ 3
𝑦 ≥ 2

Find “Corner Points” of the Feasib
Region3𝑥 + 2𝑦 ≤ 6
−2𝑥 + 4𝑦 ≤ 8
𝑥 + 𝑦 ≥ 1
𝑥 ≥ 0
𝑦 ≥ 0
𝑦 ≤ −3
2
𝑥+3
𝑦 ≤ 1
2
𝑥 + 2
𝑦 ≥ −𝑥 + 1
𝑥 ≥ 0
𝑦 ≥ 0

Find “Corner Points” of the Feasib
Region3𝑥 + 2𝑦 ≤ 6
−2𝑥 + 4𝑦 ≤ 8
𝑥 + 𝑦 ≥ 1
𝑥 ≥ 0
𝑦 ≥ 0

Finding Feasible Regions
Find the system
whose feasible region
is a triangle with
vertices (2,4),
(-4,0), and (2,-1)
2
46
832



x
yx
yx

Linear Programming
Businesses use linear
programming to find out how to
maximize profit or minimize
costs. Most have constraints on
what they can use or buy.

Linear Programming
The Objective Function is
what we need to maximize or
minimize. For us, this will be a
function of 2 variables, f(x, y)

Linear Programming
The Constraints are the
inequalities that provide us with
the Feasible Region.

The general idea… (pg 398)
Find max/min values of the objective
function, subject to the constraints.
yxyxf 52),( 
0,0
1
842
623




yx
yx
yx
yx
Objective Function Constraints

Graph the Feasible Region

The Feasible Region makes up the possible
inputs to the Objective Function
yxyxf 52),( 

Corner Point Thm (pg 400)
If a feasible region is
bounded, then the
objective function
has both a
maximum and
minimum value,
with each occurring
at one or more
corner points.

Find the minimum and maximum
value of the function f(x, y) = 3x - 2y.
We are given the constraints:
• y ≥ 2
• 1 ≤ x ≤5
• y ≤ x + 3

6
4
2
2 3 4
3
1
1
5
5
7
8
y ≤ x + 3
y ≥ 2
1 ≤ x ≤5

6
4
2
2 3 4
3
1
1
5
5
7
8
y ≤ x + 3
y ≥ 2
1 ≤ x ≤5 Need to find corner
points (vertices)

• The vertices (corners) of the
feasible region are:
(1, 2) (1, 4) (5, 2) (5, 8)
• Plug these points into the
function f(x, y) = 3x - 2y
Note: plug in BOTH x, and y values.

Evaluate the function at each vertex
to find min/max values
f(x, y) = 3x - 2y
• f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1
• f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5
• f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11
• f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

So, the optimized solution is:
• f(1, 4) = -5 minimum
• f(5, 2) = 11 maximum

Find the minimum and maximum value
of the function f(x, y) = 4x + 3y
With the constraints:
52
2
4
1
2



xy
xy
xy

6
4
2
53 4
5
1
1
2
3
y ≥ -x + 2
y ≥ 2x -5
y ≤ 1/4x + 2
Need to find corner
points (vertices)

f(x, y) = 4x + 3y
• f(0, 2) = 4(0) + 3(2) = 6
• f(4, 3) = 4(4) + 3(3) = 25
• f( , - ) = 4( ) + 3(- ) = -1 =7
3
1
3
1
3
7
3
28
3
25
3
Evaluate the function at each vertex
to find min/max values

• f(0, 2) = 6 minimum
• f(4, 3) = 25 maximum
So, the optimized solution is:

Classwork / Homework
• Page 395
•1 – 6
•7 – 45 e.o.odd
•47, 49, 53
• Page 403
•1, 3, 7, 9, 11

Lecture 7.1 to 7.2 bt

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Lecture 7.1 to 7.2 bt