systems of equations. i. systems of linear equations four methods: 1. elimination by substitution 2....

Systems of Equations

I. Systems of Linear Equations

Four Methods:

1. Elimination by Substitution

2. Elimination by Addition

3. Matrix Method

4. Cramer’s Rule

5. Geometric Method for a 2 by 2 system

Example (1)

1

,

2147

,),1()3(

)3(1024

,2)1(

lim:

1)2(25

,,

21474)25(23

,),2(

25),1(

lim:

:

)2(423

)1(52

:

y

getweEquationstheofanyinthatngSubstituti

xx

getweEquationtoEquationAdding

yx

byEquationgMultiplyin

additionbyinationEMethodSecond

y

getweEquationstheofanyinthatngSubstituti

xxxx

getweEquationinthatngSubstituti

xygetweEquationFrom

onsubstitutibyinationEdFirstMetho

Solution

yx

yx

systemfollowingtheSolve

Geometric method

Graph the lines represented by the equationsThe solution is the intersection of these lines

420-2-4

10

5

0

-5

x

y

x

y

)1,2(

Example (2)

solutionnohassystemThe

impossiblex


yx



solutionnohassystemThe

impossiblexx


xygetweEquationFrom


Solution

yx

yx


260

,),2()3(

)3(1024

,2)1(

lim:

4104)25(24

,),2(

25),1(

lim:

:

)2(424

)1(52

:

Geometric methodGraph the lines represented by the equations

These lines are parallel and do not intersect, and so no solution for the given system exists.

420-2-4

12.5

10

7.5

5

2.5

0

-2.5

-5

x

y

x

y

Example (3)

solutionsmanyekyhassystemThe

trueAlways


yx



solutionsmanyekyhassystemThe

trueAlwaysxx


xygetweEquationFrom


Solution

yx

yx


intinf

!00

,),2()3(

)3(1024

,2)1(

lim:

intinf

!101010)25(24

,),2(

25),1(

lim:

:

)2(1024

)1(52

:

Geometric method

Graph the lines represented by the two equations(they are equivalent equations) representing the same lines

420-2-4

12.5

10

7.5

5

2.5

0

-2.5x

y

x

y

3 by 3 Linear system

See the following examples:

1. Example (5) Page 144



Cramer’s Rule

Determinants

Two by Two Determinants

61218)4(3)9(294

32

Example

bcaddc

ba

Systems of Linear Equations

Two Equations in Two Unknowns

00

0

221

111

222

121

2221

12110

22221

11211

0

:

yx

yx

yandx

Then

If

ca

caand

ac

ac

aa

aa

Let

Solution

cyaxa

cyaxa

systemthesolve

Example

221

421

21

21

0

4240215

82

2132431

38

2115635

32

:

135

832

:

00

0

0

yx

y

x

yandx

Then

If

haveWe

Solution

yx

yx

systemtheSolve

The case when Δ0 = 0The left side of the first equation is a k multiple of the left side of the second one, for

some real number k

The right side of the first equation is a k

multiple of the right side of the second one

→ There are finitely many solutions for the system

The right side of the first equation is not a

k multiple of the right side of the second

one.

→ There is no solution for the system

1664

832

)1(

yx

yx

Case

1564

832

)2(

yx

yx

Case

),0()

3

)0(28,0()2,1()

3

)1(28,1(:

)3

28,(

.3

28

)2(,2)1(

2

&2

0121264

32

1664

832

)1(

38

0

orExample

solutionaisr

rpairAny

xy

uknownstwowithequationonehaveweThus

EqgetwebyEqofsidebothgMultiplyin

firsttheofsiderigtthetimesisequationfirsttheofsidelrighttThe

firsttheofsideleftthetimesisequationfirsttheofsideleftThe

Solution

yx

yx

Case

systemtheforsolutionNo

impossibleiswhich

getweEqfromequationthisgSubtractin

yx

getwebyEqofsidebothgMultiplyin

firsttheofsiderigtthetimesnotisequationfirsttheofsiderightttheBut

firsttheofsideleftthetimesisequationfirsttheofsideleftThe

Solution

yx

yx

Case

10

),2(

1664

:,2)1(

2

2

0121264

32

1564

832

)2(

0

Three by Three Determinants

54156)1(4)5(3)3(2

)32(4)61(3)41(2

23

114

13

213

12

212

123

211

432

3231

222113

3331

232112

3332

232211

333231

232221

131211

Example

aa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

A System of Three Linear Equations in Three Unknowns

000

0

33231

22221

11211

33331

23221

13111

33323

23222

13121

333231

232221

131211

0

3333231

2232221

1131211

,

0

:

zyx

z

y

x

zandyx

Then

If

caa

caa

caa

aca

aca

aca

aac

aac

aac

aaa

aaa

aaa

Let

Solution

czayaxa

czayaxa

czayaxa

systemtheSolve

Example

31

32

1

2,1

1

1

3)1(8)5(2)1(

712

611

821

2)5()1(8)1(

172

161

181

1)1()1(2)0(8

117

116

128

01)1()1(20

112

111

121

:,

72

6

82

:

0

0

zandyx

haveWe

Solution

zyx

zyx

zyx

systemtheSolve

x

z

y

x

Nonlinear SystemExample (1)

81)3(33

71)2(32

32

0)3)(2(

06

07)13(2

:),1(

13

,),2(

)2(013

)1(072

:

2

2

2

yx

oryx

xorx

xx

xx

xxx

getweEqinthatngSubstituti

xy

haveweEqFrom

Solution

yx

yxx


22

00

0)2(

02

042

04

:),1(

,),2(

:

)2(0

)1(04

:

)2(

2

2

22

22

xy

orxy

yy

yy

y

yyy

getweEqinthatngSubstituti

yx

haveweEqFrom

Solution

yx

yyx


Example

Geometric methodGraph the line and the quadratic function represented by the two

equations; y = - x2 + 2x + 7 and y = 3x + 1The points of their intersection are the solutions of the system

420-2-4

10

5

0

-5

-10

-15

x

y

x

y

)7,2(

)8,3(

Intersection of Graphs

Example (1)

22,

4)(

2)(

)(

secint

xiffexistsxthatnoticeweFirst

xxg

xxf

equationsofsystemfollowingtheSolve

functionsfollowingtheoftionertheFind

?7

)2,2())2(,2(secintint

42

7sin;2

}7,2{

2;0)7)(2(

2;0149

2;1682

2;)4(2

42

)()(

,intint

:

2

2

2

fromcomezerothedoesWhere

fistionerofpoThe

xx

equationthesatisfynotdoesxcex

x

xxx

xxx

xxxx

xxx

xx

xgxf

letwefunctionstwotheofctionserseofspothefindTo

Solution

Why we get two answers, when actually the graphs intersect at

only one point?

Answer: Because, when we squared √(x+2), we introduced

the other function whose square is also equal x+2

Which function is this?

At which point does it intersect the line y = x+2

It is the function y = - √(x+2)

It intersects the line y = x+2 at the point whose x coordinate is 7.

What is the y-coordinate of this point?

systems of equations. i. systems of linear equations four methods: 1. elimination by substitution 2....

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