Engineering Maths 3(week3).ppt

5/20/2018 Engineering Maths 3(week3).ppt

1/21

Prepared By: Charles Bong Hin Joo

KNF2033 ENGINEERING MATHEMATICS

III

LINEAR SYSTEMS OF EQUATIONS

(GAUSS ELIMINATION)


2/21

Prepared By: Dr. Charles Bong Hin Joo

Gauss Elimination


3/21


Gaussian Elimination Method

Consider a system of m equations in n unknowns given byAX=B.

Consider the augmented matrix [A l B]. Apply elementary rowoperations on both A and B such that A reduces to an uppertriangular matrix.

The n unknowns x1, x2, xnare solved by back substitutioni.e., first solve the last equation for xn, then solve the lastbut one equation for xn-1and so on.

nnnn

nn

nn

nn

bxa

bxaxa

bxaxaxa

bxaxaxaxa

***

3

*

33

*

33

2

*

23

*

232

*

22

11313212111

...................................

....

....

....


4/21



Example

624

423

9232

321

321

321

xxx

xxx

xxx


5/21



Solution

6241

4123

9232

formaugmentedIn

6

4

9

241

123

232

3

2

1

x

x

x

21205

3

5

210

6241

7

1)3(,

5

1)2(

147140

3250

6241

)1(3)3(),1(2)2(

4123

9232

6241

)3(~)2(

9232

4123

6241

)3(~)1(


6/21



41005

3

5

210

6241

)5()3(

5

4

5

100

5

3

5

210

6241

)2(2)3(

45

36

1005

210

241

3

2

1

x

xx

2684624

115

8

5

3

5

3

5

2

44

upwardsworkingrow,bottomthefromExpanding

11321

2232

33

xxxxx

xxxx

xx


7/21Prepared By: Dr. Charles Bong Hin Joo


Now you try it!!

3532

2

zyxzx

zy



Gauss Elimination (overdetermined)

Example



Gauss Elimination (underdetermined)

Example



Gauss Elimination (determined)



Gauss Elimination (no solution)



Now you try it!!

A firm produces three products, A, B and C which requireprocessing by three machines. The time (in hours) requiredfor processing one unit of each product by the threemachines is as follows:

Machine I is available for 850 hours, machine II for 1200hours and machine III for 550 hours. How many units ofeach product should be produce to make use of all theavailable time on the machine?

112

421

213

A B CMachine I

Machine II

Machine III



Echelon Form


21/21

Prepared By: Charles Bong Hin Joo

Engineering Maths 3(week3).ppt

Documents

Transcript of Engineering Maths 3(week3).ppt