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—7z=2,
= 3x
23xare
all linear
Examples.
—2x+y—z=—6,x—17z=4,4y=0,and
Examples.
3x+4y—7z=2,
—2x+y—z=—6,x—17z=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,
—1zand
is
to
the of
equation
—Zx + 5y + z = 7 since —2(1) + 5(2) 2x
+ (—1)
—2
+ 2x
5y
z+=+
2x
+=
5y5y
z+7=
7 7 1 = 7.
+
z10=
The since
point x = 3, y = —2, and z = 4 is a not a solution to the equation
since
since
—2x + 5y + z = 7 since —2(3)
+5(2)
5(—2)
(4) = 2—6
101+=47 = —12, and
2(1)
+
+
(+++
1)
2(1)
++
5(2)
( (=
1) 1)
= =+2 10
+
1010
1=
7 7
2(1)
5(2)
2+
1=
—12 $ 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.
—
—
-
—
—
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
Exercises
1.) What are the coefficients of the equation
3x
2y + 5z = 13
2.) What is the constant of the equation
3x
3.) Is x = 4, y =
2y + 5z = 13
3, and z =
1 a solution of the equation
3x
4.) Is x =
2y + 5z = 13
2, y = 0, and z = 4 a solution of the equation
x + 7y
5.) Is x = 5, y = 7, and z =
8z = 10
2 a solution of the equation
3y
5z = 31
Questions #6-10 refer to the following system of three linear equations in
three variables.
3x
4y + 2z = 9
4x + 4y +10z = 32
x + 2y 7z = 7
6.) Is x = 1, y = 4, and z = 2 a solution of the system?
7.) Is x =
1, y = 1, and z =
1 a solution of the system?
8.) Is x = 25, y = 23, and z = 4 a solution of the system?
9.) Does the system have a unique solution?
10.) Does the system have infinitely many solutions?
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For #11-13, solve the logarithmic equations for x.
11.) loge (2x
3) = 4
12.) log2 (5x + 1) = 3
13.) loge (x2 + x)
loge (x) = loge (2)
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