Linear Equations in Two Variables
In this chapter, we’ll use the geometry of lines to help us solve equations.
Linear equations in two variables.
If a, b, and r are real numbers (and if a and b are not both equal to 0) then
ax + by = r is called a linear equation in two variables. (The “two varaibles”
are the x and the y.)
The numbers a and b are called the coefficients of the equation ax + by = r.
The number r is called the constant of the equation ax + by = r.
Examples. 10x − 3y = 5 and −2x − 4y = 7 are linear equations in two
variables.
Solutions to equations.
A solution to a linear equation in two variables ax+by = r is a specific point
in R2 such that when when the x-coordinate of the point is multiplied by a,
and the y-coordinate of the point is multiplied by b, and those two numbers
are added together, the answer equals r. (There are always infinitely many
solutions to a linear equation in two variables.)
Example. Let’s look at the equation 2x − 3y = 7.
Notice that x = 5 and y = 1 is a point in R2 that is a solution to
equation because we can let x = 5 and y = 1 in the equation 2x − 3y
and then we’d have 2(5) − 3(1) = 10 − 3 = 7.
The point x = 8 and y = 3 is also a solution to the equation 2x − 3y
since 2(8) − 3(3) = 16 − 9 = 7.
The point x = 4 and y = 6 is not a solution to the equation 2x − 3y
because 2(4) − 3(6) = 8 − 18 = −10, and −10 �= 7.
this
=7
=7
=7
To get a geometric interpretation for what the set of solutions of 2x−3y = 7
looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x − 3y = 7
as 32 x− 73 = y. This is the equation of a line that has slope 23 and a y-intercept
of − 73 . In particular, the set of solutions to 2x − 3y = 7 is a straight line.
(This is why it’s called a linear equation.)
198
= aa,, so
so the
the solutions
solutions
to
this equation
equation
consist
of parallel
the
2 this
xx =
consist
of
the
in Rto
would
lie on both
of the
lines.
rr
al line
line whose
whose x-coordinates
x-coordinates equal
equal aa..
al
2.
Example. The system
x − 2y = −4
−3x + 6y
=0
ii
~
.....
(b*o)
That’s because each
of the two lines has the same
so the lines don’t intersect.
~xtr
does not have a solution.
slope,
1
2,
same ideas
ideas from
from the
the 2x
2x−
−3y
3y =
= 77 example
example that
that
we looked
looked
same
S we
that ax
ax+
+by
by =
= rr isis the
the same
same equation
equation as,
as, just
just written
written in
in
that
aa
rr
Itstraight line
m, −
−bbxx +
+ bb =
= y.
y. This
This isis the
the equation
equation of
of aa straight
line
m,
rr
nd whose
whose y-intercept
y-intercept isis bb..
nd
3
Linear
equations
and
lines.
Linear
equations
and
lines.
Linear equations
equations and
and lines.
lines.
Linear
equations
Linear
If
=
0,
then
the
linear
equation
Ifbbbbb =
= 0,
0, then
then the
the linear
linear
equation ax
ax +
+ by
by =
= rrr is
the same
same as
as ax
ax =
= r.
r.
=
0,
then
the
linear
equation
ax
+
by
=
isis the
the
same
as
ax
=
r.
+
If
=
0,
then
the
linear
equation
IfIf
rr equation ax + by = r is the same as ax = r.
r
Dividing
by
a
gives
x
=
,
so
the
solutions
to
this
equation
consist
of
the
Dividingby
byaaaagives
givesxxxx=
= araaara,,,, so
so the
the solutions
solutions to
by
gives
=
so
the
solutions
to
this
equation
consist
of
the
by
gives
=
so
Dividing
solutions
to this
this equation
equation
consist of
of the
the
Dividing
consist
Dividing
rr
rr.
points
on
the
vertical
line
whose
x-coordinates
equal
r
pointson
onthe
thevertical
verticalline
linewhose
whosex-coordinates
x-coordinatesequal
on
the
vertical
line
whose
x-coordinates
equal
on
the
vertical
line
whose
points
x-coordinates
equal aaaaa...
points
points
a
S
(b*
~xtr
2
186
*
*
*
*
*
*
*
*
*
*
*
*
*
If
�=
0,
then
the
same
ideas
from
the
2x
−
3y
=
example
that
we
looked
Ifbbbbb�=
�=0,
0,then
thenthe
thesame
sameideas
ideasfrom
fromthe
the2x
2x−
−3y
3y=
=7777example
examplethat
thatwe
welooked
looked
�=
0,
then
the
same
ideas
from
the
2x
−
3y
=
example
that
we
looked
IfIf
If
�=
0,
then
the
same
ideas
−
at
previously
shows
that
ax
+
by
=
r
is
the
same
equation
as,
just
written
in
atpreviously
previouslyshows
showsthat
thatax
ax+
+by
by=
=rrris
thesame
previously
shows
that
ax
+
by
=
isisthe
the
same
equation
as,
just
written
in
at
previously
shows
that
+
at
same equation
equation as,
as, just
just written
written in
in
at
aax
r
ax + rrrr = y. This is the equation of a straight line
a
different
form
from,
−
a
a
a
differentform
formfrom,
from,−
−bbbbxxxx+
+bbbb =
=y.
y. This
This is
is the
different
form
from,
−
+
=
y.
This
is
equation
of
straight
line
aaaadifferent
different
form
−
+
is
the equation
equation of
of aaa straight
straight line
line
afrom,
rr the
b
b
a and whose
whose
slope
is
−
y-intercept
is
.
a
r
a
r
a
r
a
whoseslope
slopeis
is−
−bbb and
and whose
whosey-intercept
y-interceptis
is bbb...
whose
slope
is
−
and
y-intercept
is
whose
whose
slope
is
−
whose
is
bbb and whose
bb
5
(b*o)
~xtr
(b*o)
~xtr
186
186
2199
2
186
Systems of linear equations.
Rather than asking for the solution set of a single linear equation in two
variables, we could take two different linear equations in two variables and
ask for all those points that are solutions to both of the linear equations.
For example, the point x = 4 and y = 1 is a solution to both of the
equations x + y = 5 and x − y = 3.
If you have more than one linear equation, it’s called a system of linear
equations, so that
x+y = 5
x−y = 3
is an example of a system of two linear equations in two variables. There are
two equations, and each equation has the same two variables: x and y.
A solution to a system of equations is a point that is a solution to each of
the equations in the system.
Example. The point x = 3 and y = 2 is a solution to the system of two
linear equations in two variables
8x + 7y = 38
3x − 5y = −1
because x = 3 and y = 2 is a solution to 3x − 5y = −1 and it is a solution to
8x + 7y = 38.
Unique solutions.
Geometrically, finding a solution to a system of two linear equations in two
variables is the same problem as finding a point in R2 that lies on each of the
straight lines corresponding to the two linear equations.
Almost all of the time, two different lines will intersect in a single point,
so in these cases, there will only be one point that is a solution to both
equations. Such a point is called the unique solution for the system of linear
equations.
Example. Let’s take a second look at the system of equations
8x + 7y = 38
3x − 5y = −1
200
The
in this system, 8x +
7y =
38, corresponds
to aa line
that
The first
first equation
8x +
+ 7y
7y =
= 38,
38, corresponds
corresponds to
to a line
line that
that
8equation in this system, 8x
has
slope
−
88. The second equation in this system, 3x−5y = 3, is represented
7
has slope − 77. The second equation
in this
this system,
system, 3x−5y
3x−5y =
= 3,
3, is
is represented
represented
3
3
by
a
line
that
has
slope
−
=
.
Since
the
two
slopes
are
not
equal, the
3
3
3
3
by a line that has slope −−5
= 555. Since
Since the
the two
two slopes
slopes are
are not
not equal,
equal, the
the
−5
−5
lines
have
to
intersect
in
exactly
one
point.
That
one
point
will
be
the
unique
lines have to intersect in exactly one point.
point. That
That one
one point
point will
will be
be the
the unique
unique
solution.
As
we’ve
seen
before
that
x
=
3
and
y
=
2
is
a
solution
to
this
solution. As we’ve seen before that xx =
= 33 and
and yy =
= 22 is
is aa solution
solution to
to this
this
system,
it
must
be
the
unique
solution.
system, it must be the unique solution.
solution.
5
Ii.
3
2.
ii
~
Example.
Example. The
The system
system
5x
= 44
5x +
+ 2y
2y
2y =
=4
−2x
+
y
=
11
−2x + yy =
= 11
11
It
has
y = 7.
has aa unique
unique solution.
solution. It’s
It’s xx =
= −2
−2 and
and
and yy =
= 7.
7.
It’s
straightforward
to
check
that
x
=
−2
and yy =
7 is aa solution
to the
It’s straightforward to3 check that xx =
= −2
−2 and
and y =
= 77 is
is a solution
solution to
to the
the
system.
That
it’s
the
only
solution
follows
from
the
fact
that
the
slope
of
the
system. That it’s the only solution follows
follows from
from the
the fact
fact that
that the
the slope
slope of
of the
the
line
5x
+
2y
=
4
is
different
from
slope
of
the
line
−2x
+
y
=
11.
Those
two
line 5x + 2y 5= 4 is 2different from slope
of
the
line
−2x
+
y
=
11.
Those
two
slope of the line −2x + y = 11. Those two
slopes
are
−
55 and 1122 respectively.
2
slopes are − 22 and 11
respectively.
11
S
No
No solutions.
solutions.
a Stwo different linear equations in two
IfIf you
you reach
reach into
into aa hat
hat and
and pull
pull out
out
out two
two different
different linear
linear equations
equations in
in two
two
variables,
it’s
highly
unlikely
that
the
two
corresponding
lines
would
have
variables, it’s highly unlikely that the
the two
two corresponding
corresponding lines
lines would
would have
have
exactly
the
same
slope.
But
if
they
did
have
the
same
slope,
then
there
exactly the same slope. But if they
did have the same slope, then there
201 did have the same slope, then there
188
4
5
Ii.
would
not
be
to
system
of
linear
equations
since
no
point
would
not
bebeaaasolution
solution
totothe
the
system
ofoftwo
two
linear
equations
since
no
point
would
not
solution
the
system
two
linear
equations
since
no
point
22
3
in
R
would
lie
on
both
of
the
parallel
lines.
ininRR2would
lie
on
both
of
the
parallel
lines.
would lie on both of the parallel lines.
2.
Example.
The
system
Example.
The
system
Example. The system
xx −
2y =
−4
x−−2y
2y==−4
−4
−3x
+
6y
=
0
−3x
==00
−3x++6y6y
ii
~
does
not
have
aa solution.
That’s
because
each
of
two
lines
has
the
same
does
not
have
solution.
That’s
because
each
ofofthe
the
two
lines
has
the
same
does
not
have
a
solution.
That’s
because
each
the
two
lines
has
the
same
11
slope,
,
so
the
lines
don’t
intersect.
slope,
22 ,1 so the lines don’t intersect.
slope, , so the lines don’t intersect.
2
S
It
3
a
S
** ** ** ** ** ** ** ** ** ** ** ** **
*
*
*
*
*
*
*
*
*
*
*
*
*
202
189
5
Exercises
1.) What are the coefficients of the equation 2x − 5y = −23 ?
2.) What is the constant of the equation 2x − 5y = −23 ?
3.) Is x = −4 and y = 3 a solution to the equation 2x − 5y = −23 ?
4.) What are the coefficients of the equation −7x + 6y = 15 ?
5.) What is the constant of the equation −7x + 6y = 15 ?
6.) Is x = 3 and y = −10 a solution to the equation −7x + 6y = 15 ?
7.) Is x = 1 and y = 0 a solution to the system
x + y=1
2x + 3y = 3
8.) Is x = −1 and y = 3 a solution to the system
7x + 2y = −1
5x − 3 y = −14
9.) What’s the slope of the line 30x − 6y = 3 ?
10.) What’s the slope of the line −10x + 5y = 4 ?
11.) Is there a unique solution to the system
30x − 6y = 3
−10x + 5y = 4
12.) What’s the slope of the line 6x + 2y = 4 ?
13.) What’s the slope of the line 15x + 5y = −7 ?
203
14.) Is there a unique solution to the system
6x + 2y = 4
15x + 5y = −7
204