2.techniques of solving algebraic equations

7/28/2019 2.Techniques of Solving Algebraic Equations

1/27

I I TECHNIQUES OF SOLVING

A L GEB RA IC EQUATIONS

Techniques of Solving Algebraic Equations

Reference : Croft, A., & Davison, R. (2008). Mathematics forEngineers - A Modern Interactive Approach, Pearson

Education.

Page 1


2/27


TYPES OF SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS

When a system of linear equations is solved, there are 3

possible outcomes:

i.

i. a unique solution

ii. an infinite number of solutions

iii. no solution

Page 2

)(1

3

1

3

10

32

10

332solutionunique

y

x

yx

yx

=

=

=+

=+


3/27


ii.

iii.

)solutionsofnumberinfinite(0

3

00

32

664

332

=+

=+

yx

yx

)(1

3

00

32

100

332solutionno

yx

yx

=+

=+

Page 3


4/27


CRAMERS RULE

Reference : Croft & Davison, Chapter 13, Blocks 1, 2

Cramers rule is a method that uses determinants to solve a

system of linear equations.

i. Two equations in 2 unknowns

If

then 0,,22

11

22

11

22

11

22

11

22

11

==ba

bathatprovided

ba

ba

ka

ka

y

ba

ba

bk

bk

x

=+

=+

222

111

kybxa

kybxa

Page 4


5/27


ii. 3 equations in 3 unknowns

If where

then

=++

=++

=++

3333

2222

1111

kzcybxa

kzcybxakzcybxa

333

222

111

333

222

111

333

222

111

333

222

111

333

222

111

333

222

111

,,

cba

cba

cba

kba

kba

kba

z

cba

cba

cba

cka

cka

cka

y

cba

cba

cba

cbk

cbk

cbk

x ===

0

333

222

111

cba

cbacba

Page 5


6/27


e.g.1 Using Cramers rule, solve for x, y.

=

=+

927

352

yx

yx

Page 6


7/27


e.g.2 Using Cramers rule, solve for x, y and z.

=+=

=++

1022312352

35435

zyxzyx

zyx

Page 7


8/27


INVERSE MATRIX METHOD

Writing System of Equations in Matrix Form

Note that

can be written as

This is called the matrix form of the simultaneous equations.

=+

=+

222

111

kybxa

kybxa

=

2

1

22

11

k

k

y

x

ba

ba

Page 8


9/27


i.e. the general matrix form of a system of equations:

where A, X and B are matrices.

AX = B

Similarly,

can also be written as

=++

=++

=++

3333

2222

1111

kzcybxa

kzcybxa

kzcybxa

=

3

2

1

333

222

111

k

k

k

z

y

x

cba

cba

cba

Page 9


10/27


Solving Equations Using the Inverse Matrix Method

Consider the matrix form: AX = BA-1AX = A-1B

I X = A-1B

X = A-1B

i.e. X can be found if A-1 exists.

Page 10


11/27


e.g.3 Redo example 1 and example 2 using the inversematrix method.

Page 11


12/27


GAUSSIAN ELIMINATION

Reference : Croft & Davison, Chapter 13, Block 3

Introduction

Gaussian Elimination is a systematic way of simplifying a

system of equations.

A matrix, called an augmented matrix, which captures all theproperties of the equations, is used.

A sequence ofelementary row operationson this matrix

eventually brings it into a form known as echelon form (to bediscussed in Page 32).

From this, the solution to the original equations is easily found.

Page 12


13/27


Augmented Matrix

Consider the system of equations,

it can be represented by an augmented matrix:

=+

=+

222

111

kybxa

kybxa

constantstscoefficien

k

k

ba

ba

2

1

22

11

this vertical line can beomitted as in your textbook

Page 13


14/27


Similarly, the following system of equations:

can also be written as an augmented matrix:

=++

=++

=++

3333

2222

1111

kzcybxa

kzcybxa

kzcybxa

3

2

1

333

222

111

k

k

k

cba

cba

cba

Page 14


15/27


e.g.1 Write down the augmented matrices for the followings

a.

b.

c.

=

=+

23127

793

yx

yx

=++

=+

=+

73224

642125

156579

zyx

zyx

zyx

=+=+

=+

753

12384

6317

zy

zyx

yx

Page 15


16/27


e.g.2 Solve the system with the augmented matrix:

a.

b.

2

15

10

71

1

1

5

100

310

121

Page 16


17/27


Row-Echelon Form of an Augmented Matrix

For a matrix to be in row-echelon form:

i. Any rows that consist entirely of zeros are the last rows ofthe matrix.

ii. For a row that is not all zeros, the first non-zero element is a

one, called a leading 1.

iii. While moving down the rows of the matrix, the leading 1s

move progressively to the right.

Page 17


18/27


e.g.1 Determine which of the following matrices are in row-

echelon form.

a. b.

22

14

78

100

810

521

27

2

5

110

100

141

Page 18


19/27


Elementary Row Operations

The elementary operations that change a system but leavethe solution unaltered are:

i. Interchange the order of the equations.

ii. Multiply or divide an equation by a non-zero constant.

iii. Add, or subtract, a multiple of one equation to, or from,

another equation.

Page 19


20/27


Note that a row of an augmented matrix corresponds to an

equation of the system of equations.

When the above elementary operations are applied to the

rows of such a matrix, they do not change the solution of the

system.

They are called elementary row operations.

Page 20


21/27


Gaussian Elimination to Solve a System of Equations

i. write down the augmented matrix.

ii. apply elementary row operations to get row-echelon form.

iii. solve the system.

Page 21


22/27

Techniques of Solving Algebraic Equations Page 22

e.g.2 Use Gaussian Elimination to solve

The augmented matrix is

Interchange row 1 and row 2

=+

=+

=++

124

433

822

zyx

zyx

zyx

1124

4331

8212

1124

8212

4331

1713140

16470

4331 row 2 2*row 1

row 3 4*row 1


23/27


15500

16470

4331row 3 2*row 2

row 2 / 7

row 3 / -5

3100

7167410

4331

Hence

The solution is

1or433

4or

7

16

7

4

3

==+

==

=

xzyx

yzy

z

{ } { }TTzyx 341=


24/27


e.g.3 Use Gaussian elimination to solve

=

=+

323

534

yx

yx

Page 24


25/27


e.g.4 Use Gaussian elimination to solve

=++

=+

=+

223

8532

1242

zyx

yxz

zyx

Page 25


26/27


Gaussian Elimination to find the Inverse of a Matrix

i. write down in a form of .

ii. apply a sequence of elementary row operations toreduce A to I.

iii. Performing this same sequence of elementary row

operations on I, we obtain A-1.

Page 26

][ IA


27/27


=

=

135

25861627

Hence

135100

25801061627001

135100

258010

2611021

135100

012210

001221

101530

012210

001221

100711

010632

001221

711

632

221

Suppose

1A

A

2.techniques of solving algebraic equations

Documents