x
+
y
=
3
→
amp
;
y
=

x
+
3
3
x
+
3
y
=
9
→
amp
;
y
=

x
+
3
That’s not a mistake; the equations are identical. What does this mean about their graphs? They lie right on top of
each other!
How many times do these lines ’intersect’? An infinite number! Every point on the line is a point of intersection.
Graphs that lie precisely on top of one another are called
coincident
. A system of coincident lines is consistent has
an infinite number of solutions. Such a system is called
dependent
(or
consistentdependent
for clarity)
In practice, consistentdependent systems appear when you are given two pieces of equivalent information.
Example D
Returning to our xylophone and yam store, the yam salesman may say "We sold 4 times more yams than xylophones
today!" while the xylophone salesman may say "We only sold onefourth the number of xylophones as we did yams
today." We can see that these two statements are saying the same thing in two different ways, and they will produce
equivalent equations:
y
=
4
x
→
amp
;
y
=
4
x
x
=
1
4
y
→
amp
;
y
=
4
x
Example E
Determine if the system is consistent, inconsistent, or consistentdependent
3
x

2
y
=
4
9
x

6
y
=
1
231
4.2. Types of Linear Systems
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Fist, put the equations into slopeintercept form:
3
x

2
y
=
4
→
amp
;
y
=
3
2
x

2
9
x

6
y
=
1
→
amp
;
y
=
3
2
x

1
6
We see that these lines have the same slope but different yintercepts. Therefore:
• These lines are parallel.
• The system has no solution.
• The system is inconsistent.
Summary
There are three possibilities for the solutions to a linear system:
• One solution  Consistent and Independent
• No solutions  Inconsistent
• An infinite number of solutions  Consistent and dependent
Example F
Two movie rental websites are in competition. Blamazon charges an annual membership of $60 and charges $3 per
movie rental, while Nitflex charges an annual membership of $40 and charges $3 per movie rental. After how many
movie rentals would Blamazon become the better option?
It should already be clear to see that Blamazon will never become the better option, since its membership is more
expensive and it charges the same amount per movie as Nitflex.
Let’s see how this works algebraically.
Define the variables: Let
x
=
number of movies rented and
y
=
total rental cost
y
=
60
+
3
x
Blamazon
y
=
40
+
3
x
Nitflex
The lines that describe each option have different
y

intercepts, namely 60 for Blamazon and 40 for Nitflex. They
have the same slope, three dollars per movie. This means that the lines are parallel and the system is inconsistent.
The system has no solutions, so there is no number of rentals for which the two websites will have the same cost.
MEDIA
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URL:
232
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Chapter 4.
Unit 5  Systems of Equations
MEDIA
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