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 variables”
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
variables.
3y = 5 and
2x
4y = 7 are linear equations in two
Solutions of equations
A solution of 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 of
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 of the equation 2x 3y
since 2(8) 3(3) = 16 9 = 7.
The point x = 4 and y = 6 is not a solution of the equation 2x 3y
because 2(4) 3(6) = 8 18 = 10, and 10 6= 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 23 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.)
244
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H
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Linear
equations
and lines.
lines
LinearLinear
equations
and lines.
Linear
equations
and
lines.
equations
and
If bthen
= 0,
0,
then
theequation
linear equation
equation
ax
+
by
= rrrsame
is the
theas
same
asr.
ax =
= r.
r.
If b = If
0,
the
linear
ax +
=
rrby
is
the
ax
+ by
byax
=+
is =
the
same
assame
ax =
=
r.
If
bb =
=
0,
then
the
linear
equation
ax
+
by
=
the
same
as
ax
=
r.
then
the
linear
isis
as
ax
r
r
r
rr ,solutions
Dividing
by aaxa gives
gives
=
so the
the solutions
solutions
of this
this equation
equation
consist
of the
the
DividingDividing
by a gives
=
so
the
to
consist
of
to this
this equation
equation
consist consist
of the
the of
Dividing
by
gives
xx =
=
, solutions
so
to
this
equation
consist
of
the
by
aa, x
aaa, so the solutions to
r
r
points
onthe
thevertical
vertical
linex-coordinates
whosex-coordinates
x-coordinates
equal rra...
points on
the vertical
line
whose
equal
x-coordinates
equal aar.equal
.equal
points
on
the
vertical
line
whose
x-coordinates
points
on
line
whose
aa
(b*
~xtr
Ifthen
6=0,
0,then
then
the
same
ideas
from
the
2x
3y=
=777example
example
that
welooked
looked
If b ⇥=
the
same
ideas
from
thefrom
2x the
3y
=
that
looked
6 0,IfIf
3y2x
= 77 example
example
that we
wethat
looked
6=
0,
then
the
same
ideas
from
the
2x
3y
=
example
that
we
looked
bbb⇥=
the
same
ideas
3y
we
atpreviously
previously
shows
that
ax+
+is
bythe
=rrsame
rsame
isthe
the
sameequation
equation
as,
justwritten
written
in
at previously
shows that
axthat
+
by ax
=
r+
equation
as,
written
in
equation
as, just
justas,
written
in
at
previously
shows
that
ax
by
=
the
same
equation
as,
just
written
in
at
shows
by
=
isis
same
just
in
a
r
a
r
a
r aa xy.+ rThis
di↵erent
form from,
from,
= y.
y.
Thisequation
is the
the equation
equation
of aaa straight
straight
line
a different
form from,
is
the
of
line
is This
the
equation
of aa straight
straight
line line
different
form
from,
x+
+ brbb =
=
y.
This
the
equation
of
straight
line
aaa different
form
isis
of
bx
bab x + bb =
b
b
r
a
r
r
a
r
a
r
whose
slope
is whose
andy-intercept
whose y-intercept
y-intercept
is b...
whose slope
is slope
is
is bb.. isis
whose
slope
and
whose
whose
isis
b and
bbb and whose y-intercept
bb
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2
186
oj
(b*o)
~xtr
Systems of linear equations
Rather than asking for the set of solutions of a single linear equation in two
variables, we could take two di↵erent 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 of 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 of a system of equations is a point that is a solution of each of
the equations in the system.
Example. The point x = 3 and y = 2 is a solution of the system of two
linear equations in two variables
8x + 7y = 38
3x
5y = 1
because x = 3 and y = 2 is a solution of 3x
8x + 7y = 38.
5y =
1 and it is a solution of
Unique solutions
Geometrically, finding a solution of 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 di↵erent 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 of the system of linear
equations.
Example. Let’s take a second look at the system of equations
8x + 7y = 38
3x
5y = 1
246
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
3x 5y
5y =
= 3,
3, is
is represented
represented
3
3
by
a
line
that
has
slope
353 = 533. Since the two slopes are not equal, the
by a line that has slope
Since the
the two
two slopes
slopes are
are not
not equal,
equal, the
the
55 = 55. Since
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,
x
=
3
and
y
=
2
is
a
solution
of
this
system.
solution. As we’ve seen before that xx =
= 33 and
and yy =
= 22 is
is aa solution
solution to
to this
this
It
is
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 =
= 22 and
and
and yy =
= 7.
7.
It’s
straightforward
to
check
that
x
=
2
and yy =
7 is aa solution
of the
It’s straightforward to3 check that xx =
= 22 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
di↵erent
from
slope
of
the
line
2x
+
y
=
11.
Those
two
line 5x + 2y 5= 4 is different from slope
of
the
line
2x
+
y
=
11.
Those
two
slope of the line 2x + y = 11. Those two
slopes
are
55 and 222respectively.
2
slopes are 22 and 11
respectively.
11
S
No
No solutions
solutions.
a Stwo di↵erent 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
247 did have the same slope, then there
188
4
5
Ii.
would
not
be
of
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== 44
3x
+
3x
++6y
6y6y=
==000
3x
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
** ** ** ** ** ** ** ** ** ** ** ** **
*
*
*
*
*
*
*
*
*
*
*
*
*
248
189
5
Exercises
1.) What are the coefficients of the equation 2x
2.) What is the constant of the equation 2x
3.) Is x =
5y =
4 and y = 3 a solution of the equation 2x
4.) What are the coefficients of the equation
5.) What is the constant of the equation
6.) Is x = 3 and y =
x + y=1
2x + 3y = 3
1 and y = 3 a solution of the system
9.) What’s the slope of the line 30x
10.) What’s the slope of the line
1
14
6y = 3 ?
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 =
249
23 ?
5y =
23 ?
7x + 6y = 15 ?
10 a solution of the equation
7x + 2y =
5x
3y=
23 ?
7x + 6y = 15 ?
7.) Is x = 1 and y = 0 a solution of the system
8.) Is x =
5y =
7?
7x + 6y = 15 ?
14.) Is there a unique solution to the system
6x + 2y = 4
15x + 5y = 7
For #15-17, find the roots of the given quadratic polynomials.
15.) 9x2
12x + 4
16.) 2x2
3x + 1
17.)
4x2 + 2x
3
250