Example:
`
1
:
4
x
+ 6
y
= 6
,
y
=

2
3
x
+ 1
`
2
:
2
x
+ 3
y
= 5
,
y
=

2
3
x
+
5
3
Math 17
(UPIMath)
Systems
Lec 12
15 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..are parallel lines, then the system has no solution and is said to be
inconsistent
.
Example:
`
1
:
4
x
+ 6
y
= 6
,
y
=

2
3
x
+ 1
`
2
:
2
x
+ 3
y
= 5
,
y
=

2
3
x
+
5
3
9
>
>
=
>
>
;
Parallel.
Math 17
(UPIMath)
Systems
Lec 12
15 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..coincide, then there are infinitely many solutions to the system. The
system is said to be
dependent
.
Math 17
(UPIMath)
Systems
Lec 12
16 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..coincide, then there are infinitely many solutions to the system. The
system is said to be
dependent
.
Ex.
`
1
:
4
x
+ 6
y
= 6
`
2
:
2
x
+ 3
y
= 3
Math 17
(UPIMath)
Systems
Lec 12
16 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..coincide, then there are infinitely many solutions to the system. The
system is said to be
dependent
.
Ex.
`
1
:
4
x
+ 6
y
= 6
,
y
=

2
3
x
+ 1
`
2
:
2
x
+ 3
y
= 3
,
y
=

2
3
x
+ 1
Math 17
(UPIMath)
Systems
Lec 12
16 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..coincide, then there are infinitely many solutions to the system. The
system is said to be
dependent
.
Ex.
`
1
:
4
x
+ 6
y
= 6
,
y
=

2
3
x
+ 1
`
2
:
2
x
+ 3
y
= 3
,
y
=

2
3
x
+ 1
9
>
>
=
>
>
;
Same line.
Math 17
(UPIMath)
Systems
Lec 12
16 / 29
Two Lines
In a system of two linear equations in two variables, if the lines
representing the equations..
..coincide, then there are infinitely many solutions to the system. The
system is said to be
dependent
.
Ex.
`
1
:
4
x
+ 6
y
= 6
,
y
=

2
3
x
+ 1
`
2
:
2
x
+ 3
y
= 3
,
y
=

2
3
x
+ 1
9
>
>
=
>
>
;
Same line.
Note: A system of two linear equations is
dependent
if one equation can
be obtained by multiplying both sides of the other by a nonzero constant.
Math 17
(UPIMath)
Systems
Lec 12
16 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
Math 17
(UPIMath)
Systems
Lec 12
17 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
Solution: By Substitution Method,
Math 17
(UPIMath)
Systems
Lec 12
17 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
Solution: By Substitution Method,
2
x

3
=
x
2

4
x
+ 2
(Take value of
y
in
`
and replace
y
in
p
)
Math 17
(UPIMath)
Systems
Lec 12
17 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
Solution: By Substitution Method,
2
x

3
=
x
2

4
x
+ 2
(Take value of
y
in
`
and replace
y
in
p
)
)
0
=
x
2

6
x
+ 5
(Solve for
x
)
Math 17
(UPIMath)
Systems
Lec 12
17 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
Solution: By Substitution Method,
2
x

3
=
x
2

4
x
+ 2
(Take value of
y
in
`
and replace
y
in
p
)
)
0
=
x
2

6
x
+ 5
(Solve for
x
)
0
=
(
x

1)(
x

5)
Math 17
(UPIMath)
Systems
Lec 12
17 / 29
A Line and a Parabola
Ex. Find the points of intersection of the graphs of
p
:
y
=
x
2

4
x
+ 2
and
`
:
y
= 2
x

3
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 Spring '11
 AaronJamesPorlante
 Math, Linear Equations, Systems Of Equations, Equations, Quadratic equation, Elementary algebra, Interpretation of Linear Systems