Slide 1 © 2005 Baylor University

Fundamentals of Engineering AnalysisEGR 1302 Unit 1, Lecture G

Approximate Running Time - 20 minutesDistance Learning / Online Instructional Presentation

Presented byDepartment of Mechanical Engineering

Baylor University

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Slide 2 © 2005 Baylor University

22

533

3

321

321

321

xxx

xxx

xxx

2211

5313

3111

1320

4640

3111

13'

3

12'

2 3

rrr

rrr

23'

3 2

1rrr

1000

4640

3111

Compatible and Incompatible Solutions

?1*0 3 ximplies

Note: detA=0

This system has no solution“inconsistent”

Slide 3 © 2005 Baylor University

2211

5313

1111

2

1

2

3,1)

2

3

2

1(

2

3

2

1,264

11

22

3

xx

xx

x

1320

2640

1111

13'

3

12'

2 3

rrr

rrr

0000

2640

1111

23'

3 2

1rrr

2

3

2

12

1

2

3

3

2

1

x

x

x

The Infinite Solution

Change the System

Same Rowoperations

This system has infinite solutions, depending on the value of

0*0 3 ximplies

detA still equals zero

Slide 4 © 2005 Baylor University

The Three General Solutions

k100

10

1

1. Unique = X3 exists as a single value0det A

k000

10

1

2. None = No X3 exists0det A

0000

10

1

3. Infinite = X3 exists as any value0det A

Slide 5 © 2005 Baylor University

Graphic Examples of the Three General Solutions

X Y S( ) Xt Yt P( ) L M N( )

X Y S( ) Xt Yt P( ) L M N( )

X Y S( ) Xt Yt P( ) L M N( )

12734

56312

102102

zyx

zyx

zyx

22

533

1

zyx

zyx

zyx

22

2233

1

zyx

zyx

zyx

585809

585103

585571

100

010

001

0000

10

01

21

23

23

21

1000

010

001

23

21

rref()

rref()

rref()

infinite

unique

no solution

planesparallel,

neverintersect

all planesintersect

on the same line

single pointof intersection

Slide 6 © 2005 Baylor University

0*3

02

0

321

21

321

xxxx

xx

xxx

031

0021

0111

x

0500

0110

0111

x

5det xA

0xAThe Homogeneous Set of Linear Equations

2,0

,0

11

22

3

xx

xx

x

A

cc

bb

aa

xdet

0

0

0

21

21

21

Using Cramer’s rule:

This determinant = 0,

implying 0*det xA

0xEither , a trivial solution, or

0det A , implying infinite solutions

When x=5, infinite solutions exist,otherwise, there is no solution.

rref()

Slide 7 © 2005 Baylor University

Example of the Three General Solutions(Example 12.3 in the Text)

1a1

1

a

bz

ba11

3212

2321

Given13

'3

12'

2 2

rrr

rrr

23

'3 rrr &

1100

1430

2321

ba

)1(3det aA

1. Unique Solution

1b1a3. Infinite Solutions0

0z

1b1a2. No Solutions0

kz

Slide 8 © 2005 Baylor University

This concludes Unit 1, Lecture G

You are now ready to take the Unit 1 Exam