2.techniques of solving algebraic equations
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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.
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Techniques of Solving Algebraic Equations
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
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)(1
3
1
3
10
32
10
332solutionunique
y
x
yx
yx
=
=
=+
=+
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Techniques of Solving Algebraic Equations
ii.
iii.
)solutionsofnumberinfinite(0
3
00
32
664
332
=+
=+
yx
yx
)(1
3
00
32
100
332solutionno
yx
yx
=+
=+
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
e.g.1 Using Cramers rule, solve for x, y.
=
=+
927
352
yx
yx
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Techniques of Solving Algebraic Equations
e.g.2 Using Cramers rule, solve for x, y and z.
=+=
=++
1022312352
35435
zyxzyx
zyx
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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.
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Techniques of Solving Algebraic Equations
e.g.3 Redo example 1 and example 2 using the inversematrix method.
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Techniques of Solving Algebraic Equations
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.
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
e.g.2 Solve the system with the augmented matrix:
a.
b.
2
15
10
71
1
1
5
100
310
121
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Techniques of Solving Algebraic Equations
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.
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Techniques of Solving Algebraic Equations
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
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Techniques of Solving Algebraic Equations
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.
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Techniques of Solving Algebraic Equations
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.
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Techniques of Solving Algebraic Equations
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.
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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
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Techniques of Solving Algebraic Equations Page 23
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=
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Techniques of Solving Algebraic Equations
e.g.3 Use Gaussian elimination to solve
=
=+
323
534
yx
yx
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Techniques of Solving Algebraic Equations
e.g.4 Use Gaussian elimination to solve
=++
=+
=+
223
8532
1242
zyx
yxz
zyx
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Techniques of Solving Algebraic Equations
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.
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][ IA
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Techniques of Solving Algebraic Equations Page 27
=
=
135
25861627
Hence
135100
25801061627001
135100
258010
2611021
135100
012210
001221
101530
012210
001221
100711
010632
001221
711
632
221
Suppose
1A
A