Linear Equations in Three Variables - University of Utah

Linear Equations in Three Variables

Linear

Equationsinin

inThree

ThreeVariables

Variables

Linear

Equations

Three

Variables

Linear

Equations

Variables

2

R is the space of 2 dimensions. There is an x-coordinate that can be any

is the

the

space

dimensions.

There

isan

an

x-coordinate

that

canbe

beany

any

RR

is

space

22dimensions.

There

is

x-coordinate

that

can

JR2

isis22the

space

of

22of

dimensions.

There

isis an

x-coordiuatu

number,

and

there

is a y-coordinate

that

can

be any IJIHI

real

number.

JR2real

the

space

of of

dimensions.

There

an

x-coordiuatu

IJIHI

3

real

number,

and

there

y-coordinate

that

can

be

any

number. Each

real

and

there

aay-coordinate

that

can

any

real

number.

real

number,

there

ais

that

be

any

real

number.

Rnumber,

is theand

space

of is3is dimensions.

There

iscan

an

x,be

y,

and

zreal

coordinate.

real

number,

and

there

aisy-coordinate

y-coordinate

that

can

be

any

real

number.

33

is the

the

space

dimensions.

There

isan

an

x,y,

y,and

and

z coordinate.

coordinate.

Each

RR

is

space

33 dimensions.

There

isan

zcoordinate.

Each

R3

isis the

space

of

33ofany

dimensions.

There

isis an

.x,

y,

and

zz coordinate.

Each

can

real number.

R3coordinate

the

space

ofbeof

dimensions.

There

.x, x,

y,

and

Each

coordinate

can

be

any

real

number.

coordinate

any

real

number.

coordinate

can

be

any

real

number.

coordinate

cancan

bebe

any

real

number.

:2.:2.

II

Linear

Linear equations

equations in

in three

three variables

variables

IfIf a,a, b,b, ccequations

and

numbers

ififa,a,b,b,and

Linear

three(and

variables

and rr are

are real

realin

numbers

(and

andccare

arenot

notall

allequal

equalto

to0)

0)

then

ax

++cz

==in

rreal

called

avariables.

variables.

then

ax

by

r isthree

is

calledvariables.

a linear

linearifequation

equation

inc three

threenot

variables.

(The

Linear

equations

Linear

equations

in

three

If a,

b,++cby

and

rczare

numbers

(and

a, b, andin

are

all equal(The

to 0)

are

x,

the

z.)

¡°three

are

the

x,called

they,y,

and

the

then

+ byrr +

=the

rnumbers

is

aand

linear

equation

in not

three

(The

If

a,

cvariables¡±

and

are

real

(and

ififthe

a,

equal

If¡°three

a, b,b, ax

cvariables¡±

and

arecz

real

numbers

(and

a, b,

b,z.)and

and cc are

are

not all

allvariables.

equal to

to 0)

0)

The

b,

and

called

the

ofof the

The

The

numbers

and

caaare

are

called

the

coefficients

the equation.

equation.

The

variables¡±

the cx,

the

y, and

thecoefficients

z.)

then

ax

by

+

rr are

is

linear

equation

in

(TJir¡¯

then¡°three

ax +

+numbers

by

+ ez

ez ==a,a,

isb,called

called

linear

equation

in two

two variables.

variables.

(TJir¡¯ ¡°tliv¡¯~

¡°tliv¡¯~

number

rare

called

the

constant

the

numberare

r isisthe

called

they,

constant

theequation.

equation.

variables¡±

x,

and

z.)

variables¡±

the

x, the

the

y,

and the

theofof

z.)

Examples. 3x + 4y 7z = 2, 2x + y z = 6, x 17z = 4, 4y = 0, and

Examples.

++4y

7z

2,

zz== variables.

6,6,xx 17z

Examples.

4y

7z==equations

2, 2x

2x++yin

y three

17z==4,4,4y

4y==0,0,and

and

x + y + z 3x+4y¡ª7z=2,

= 3x

23xare

all linear

Examples.

¡ª2x+y¡ªz=¡ª6,x¡ª17z=4,4y=0,and

Examples.

3x+4y¡ª7z=2,

¡ª2x+y¡ªz=¡ª6,x¡ª17z=4,4y=0,and

22are

linear

equations

variables.

are

all

linear

equations

inthree

three

variables.

xx ++xxyy++++yyzz++==zz2=

all

linear

equations

in

three

variables.

2=are

are

allall

linear

equations

in in

three

variables.

Solutions to equations

Solutions

Solutions

to

equations

Solutions

to

equations

Solutions

toto

equations

A solution

to

a equations

linear

equation in three variables ax+by+cz = r is a specific

3 to

AA solution

oflinear

linear

equation

three

variables

+ point

by

+isrcz

=

r is a

solution

aa linear

equation

inin

three

variables

ax+by+cz

=

a specific

in Rto

that

when

when

the

x-coordinate

ofaxthe

is

multiplied

A

solution

asuch

equation

in

three

variables

ax+by+cz

=

aaisspecific

Apoint

solution

to

a

linear

equation

in

three

variables

ax+by+cz

= rr is

specific

3

3 in R such that when

specific

of

point

is

point

inpoint

R

such

when

x-coordinate

the

point

ismultiplied

multiplied

byin

a,JR3

the

y-coordinate

ofwhen

thewhen

point

is multiplied

by

b, the

z-coordinate

of the

point

such

that

when

the

x-coordinate

of

point

isis multiplied

point

in

JR3

such

thatthat

when

when

thethe

x-coordinate

of the

the

point

multiplied

by

a,

the

y-coordinate

of

the

point

is

multiplied

by

b,

the

z-coordinate

of

the

by

a,

the

y-coordinate

of

the

point

is

multiplied

by

b,

the

z-coordinate

of

the

point

multiplied by

c,

then

threeby

products

are

added

together,

by

the

y-coordinate

of

point

isis those

multiplied

b,b, and

by a,

a,

the is

y-coordinate

of the

theand

point

multiplied

by

and the

the z-coordinate

z-coordinate

point

isis multiplied

by

c,

then

those

three

products

added

together,

point

multiplied

c, and

and

then

those

three

products

are

added

together,

thepoint

answer

equals

r.by(There

are

always

infinitely

many

solutions

to

a linear

of

isis multiplied

by

c,c, and

those

three

numbers

are

added

together,

of the

the

point

multiplied

by

and

those

three

numbers

areare

added

together,

the

answer

equals

r.

(There

are

always

infinitely

many

solutions

to

a

answer

r. (There

always

infinitely

many

solutions

alinear

linear

equation

in equals

three

the

answer

equals

r.r. variables.)

(There

are

always

infinitely

many

solutions

to

the the

answer

equals

(There

are are

always

infinitely

many

solutions

to ato

a linear

linear

1

equation

in

variables.)

equation

inthree

three

variables.)

equation

in

variables.)

equation

in three

three

variables.)

255

192

11

Example.

The

xy =x

1,

y1,

=

and

=z a=

1=is1 ais1solution

the

equation

Example.

The

point

=

y 2,

=

2,

and

ais solution

to to

the

equation

Example.

The

point

1,

yz =

zsolution

a solution

the

equation

Example.

The point

xpoint

= 1,

= x

2, =

and

= 2,

¡ª1zand

is

to

the of

equation

¡ªZx + 5y + z = 7 since ¡ª2(1) + 5(2) 2x

+ (¡ª1)

¡ª2

+ 2x

5y

z+=+

2x

+=

5y5y

z+7=

7 7 1 = 7.

+

z10=

The since

point x = 3, y = ¡ª2, and z = 4 is a not a solution to the equation

since

since

¡ª2x + 5y + z = 7 since ¡ª2(3)

+5(2)

5(¡ª2)

(4) = 2¡ª6

101+=47 = ¡ª12, and

2(1)

+

+

(+++

1)

2(1)

++

5(2)

( (=

1) 1)

= =+2 10

+

1010

1=

7 7

2(1)

5(2)

2+

1=

¡ª12 $ 7.

The

point

x=

y =

and

zand

=

4z is

solution

of to

theto

equation

The

point

x 3,

==

3,

y =

and

z =

4 ais

a not

a solution

the

equation

The

point

x

3,

y 2,

= 2,

2,

=

4 not

is

a anot

a solution

the

equation

+ 2x

5y

+ 5y

z+=+

since

2x

++

5y

z 7=

7 since

zand

=

7 since

Linear2xequations

planes.

The set of solutions in2(3)

R2

to5(

linear

in6 10

two

+

=(4)

+

4+=variab1r¡¯~

2(3)

+a+

5(2)

2) (4)

+equation

(4)

=6= 6 10

4=

2(3)

5(+

2)

+

10

+

412

= 1212 1 1

dimensional

line.

andand

and

The set of solutions in F to a linear equation

in three variables is a 212 12

6= 12

7?= 6=

7 7

dimensional plane.

A solution

to equations

a linear

equation

inand

three

variables ax + by + cz = r is

Linear

and

planes

Linear

equations

and

planes

Linear

equations

planes

a point in

R3set

that

lies

on the inplane

corresponding

to ax +

+ cz

= r. So

2 2a linear equation

The

ofset

solutions

R2inRof

in by

two

variables

is ais 1The

of

in

two

variables

a 1Theset

ofsolutions

solutionsin

Rto toa linear

a linearequation

equation

in

two

variables

is

a 1we seedimensional

that there are

more

solutions

to

a

linear

eqnation

in

three

variables

line.

dimensional

line.

dimensional

line.

(2-dimensions

worth)

then

there

are

to

a linear

eqnation

in twovariables

variablesis a 2The

setset

ofset

solutions

in in

R3in

a3 to

linear

equation

in in

three

The

of

solutions

Rof3Rto

a linear

equation

three

variables

is is

a 2The

of

solutions

a

linear

equation

in

three

variables

a 2(1-dimensions

worth)

dimensional

plane.

dimensional

plane.

dimensional

plane.

¡ª

¡ª

-

¡ª

¡ª

ax +

C I ¡®C

Solutions

of to

systems

of of

linear

equations

Solutions

equations

Solutions

tosystems

systems

oflinear

linear

equations

As As

inAs

the

previous

chapter,

we we

can

have

a system

of linear

equations,

andand

in in

the

previous

chapter,

can

have

a system

of of

linear

equations,

Solutions

to

systems

of

linear

equations.

the

previous

chapter,

we

can

have

a system

linear

equations,

and

we we

can

try

totry

find

solutions

that are

common

to to

each

of the

equations

in thethethe

can

try

to

find

solutions

are

common

each

of of

the

equations

we

can

to

find

solutions

that

common

to

each

the

equations

As in the

previous

chapter,

we can that

have

aare

system

of linear

equations,

and in in

system.

system. solution to each equation in the system.

ask for a system.

common

WeWe

callcall

acall

solution

to to

a to

system

of of

equations

unique

if there

areare

no

other

a solution

a of

system

equations

unique

if there

no

other

a tosolution

a system

of

equations

unique

if there

are

no

other

If there is aWe

solution

a system

equations,

we call that

solution

unique

solutions.

solutions.

if theresolutions.

are

no other solutions.

256

9

193 2

Example.

The

point

and

solution

to

the

following

Example.

Example. The

Thepoint

pointxxx==

=3,3,

3,yyy==

=0,0,

0,and

andzzz==

=111isis

isaaasolution

solutionto

ofthe

thefollowing

following

system

of

three

linear

equations

in

three

variables

system

of

three

linear

equations

in

three

variables

system of three linear equations in three variables

3x

2y

5z

14

3x

3x++

+2y

2y 5z

5z==

= 14

14

2x

3y

4z

10

2x

2x 3y

3y++

+4z

4z==

=10

10

xxx++

+ yyy ++

+ zzz==

=444

That¡¯s

because

we

can

substitute

and

for

x,

and

respectively

in

That¡¯s

That¡¯sbecause

becausewe

wecan

cansubstitute

substitute3,3,

3,0,0,

0,and

and111for

forx,

x,y,y,

y,and

andzzz respectively

respectivelyin

in

the

equations

above

and

check

that

the

theequations

equationsabove

aboveand

andcheck

checkthat

that

3(3)

2(0)

5(1)

14

3(3)

3(3)++

+2(0)

2(0) 5(1)

5(1)==

= 999 555==

= 14

14

2(3)

3(0)

4(1)

10

2(3)

2(3) 3(0)

3(0)++

+4(1)

4(1)==

=666++

+444==

=10

10

(3)

(0)

(1)

(3)

(3)++

+ (0)

(0) ++

+ (1)

(1)==

=333++

+111==

=444

Geometry

of

solutions

Geometry

Geometry of

of solutions

solutions

Suppose

you

have

system

of

three

linear

equations

in

three

variables.

Suppose

Suppose you

you have

have aaa system

system of

of three

three linear

linear equations

equations in

in three

three variables.

variables.

33

Each

of

the

three

equations

has

a

set

of

solutions

that¡¯s

a

plane

in

Each

Each ofof the

the three

three equations

equations has

has aa set

set ofof solutions

solutions that¡¯s

that¡¯s aa plane

plane inin RR

R3... AA

A

solution

to

the

system

of

equations

is

a

point

that

lies

on

all

three

of

those

solution

solutiontoof the

the system

system ofof equations

equations isis aa point

point that

that lies

lies on

on all

allthree

threeofofthose

those

planes.

there

only

one

point

that

lies

on

all

three

planes,

then

that

planes.

planes. IfIf

If there

there isis

is only

only one

one point

point that

that lies

lies on

on all

all three

three planes,

planes, then

then that

that

solution

is

unique.

solution

is

unique.

solution is unique.

you

randomly

write

down

three

different

linear

equations

in

three

variIfIf

Ifyou

yourandomly

randomlywrite

writedown

downthree

threedifferent

di?erentlinear

linearequations

equationsin

inthree

threevarivariables,

the

odds

are

that

the

three

corresponding

planes

will

intersect

in

one,

ables,

ables,the

theodds

oddsare

arethat

thatthe

thethree

threecorresponding

correspondingplanes

planeswill

willintersect

intersectininone,

one,

and

only

one,

point.

That

means

that

for

most

systems

three

linear

equaand

andonly

onlyone,

one,point.

point. That

Thatmeans

meansthat

thatfor

formost

mostsystems

systemsofof

ofthree

threelinear

linearequaequations

in

three

variables,

there

will

be

a

unique

solution.

tions

in

three

variables,

there

will

be

a

unique

solution.

tions in three variables, there will be a unique solution.

3

194

257

mightbebethat

thatthe

thethree

threeplanes

planesfrom

fromthe

thesystem

systemofofthree

threeequations

equationswill

will

ItItmight

It

might

be

that

the

three

planes

from

the

system

of

three

equations

will

be

parallel.

Then

thethree

threeplanes

planesfrom

wouldn¡¯t

intersect.

There¡¯d

point

Itparallel.

might

beThen

thatthe

the

three

planes

from

thesystem

system

three

equations

will

It

might

be

that

the

ofofthree

equations

will

be

planes

wouldn¡¯t

intersect.

There¡¯d

bebenonopoint

be

parallel.

Then

the

three

planes

wouldn¡¯t

intersect.

There¡¯d

be

no

point

common

allthree

three

planes,

andhence

hence

thesystem

system

willnot

not

haveany

any

solutions.

be

parallel.

Then

the

threeplanes

planes

wouldn¡¯t

intersect.

There¡¯d

besolutions.

no

point

be

parallel.

Then

the

three

wouldn¡¯t

intersect.

There¡¯d

be

no

point

common

totoall

planes,

and

the

will

have

common

to

all

three

planes,

and

hence

the

system

will

not

have

any

solutions.

commonto

toall

allthree

threeplanes,

planes,and

andhence

hencethe

thesystem

systemwill

willnot

nothave

haveany

anysolutions.

solutions.

common

\¡®

4¡¯

¡®S

VS

Theremight

mightnot

notbebea apoint

pointthat

thatlies

lieson

onall

allthree

threeplanes

planeseven

evenififthe

theplanes

planes

There

There

might

not

be

a

point

that

lies

on

all

three

planes

even

if

the

planes

Theremight

mightnot

not

be

pointagain,

thatlies

lies

onall

all

three

planeseven

even

theplanes

planes

aren¡¯t

parallel.

Inbe

this

case

there¡¯d

bethree

no solution

at

all.

There

aapoint

that

on

planes

ififthe

aren¡¯t

aren¡¯t parallel.

parallel. In

In this

this case

case again,

again, there¡¯d

there¡¯d be

be no

no solution

solution at

at all.

all.

aren¡¯tparallel.

parallel. In

Inthis

thiscase

caseagain,

again,there¡¯d

there¡¯dbe

beno

nosolution

solutionatatall.

all.

aren¡¯t

1954

258

1954

Sometimes, thethree

three planeswill

will intersectinina away

way thatallows

allows formore

more

Sometimes,

Sometimes,the

the threeplanes

planes willintersect

intersect in a waythat

that allowsfor

for more

than onepoint

point to be onallallthree

three planesatatonce.

once. In thiscase,

case, thereare

are

than

thanone

one pointtotobebeon

on all threeplanes

planes at once. InInthis

this case,there

there are

multiple solutions. Because

Because there¡¯smore

more thanone

one solution,there¡¯s

there¡¯s nota a

multiple

multiplesolutions.

solutions. Becausethere¡¯s

there¡¯s morethan

than onesolution,

solution, there¡¯snot

not a

uniquesolution.

solution.

unique

unique solution.

A

3 3 and try to convince

Visualize di?erentarrangements

arrangements of threeplanes

planes in R

Visualize

Visualizedifferent

different arrangementsofofthree

three planesininRR3and

andtry

trytotoconvince

convince

yourself

that

either

there

is

exactly

one

point

contained

in

all

three

planes,

yourself

yourselfthat

thateither

eitherthere

thereisisexactly

exactlyone

onepoint

pointcontained

containedininallallthree

threeplanes,

planes,

ornonopoints

points containedininallallthree

three planes,ororthat

that thereare

are infinitelymany

many

oror

no pointscontained

contained in all threeplanes,

planes, or thatthere

there areinfinitely

infinitely many

points thatare

are containedininallallthree

three planes. That

That meansthat

that a system

points

pointsthat

that arecontained

contained in all threeplanes.

planes. Thatmeans

means thata asystem

system

ofthree

threelinear

linearequations

equationsininthree

threevariables

variableswill

willalways

alwayshave

haveeither

eithera aunique

unique

ofof

three linear equations in three variables will always have either a unique

solution, nosolution

solution at all,ororinfinitely

infinitely manysolutions.

solutions.

solution,

solution,no

no solutionatatall,

all, or infinitelymany

many solutions.

*** *** *** *** *** *** * ** * ** * ** * ** * ** * ** * **

259

1965

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