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Section 4.1
Solving Systems
of Equations in
Two Variables by
Graphing
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
Systems of Equations
A system of equations or system of inequalities is
two or more equations or inequalities in several
variables that are considered simultaneously.
y
y
y
3
2
3
2
3
2
1
1
1
x
3 2 1
1
2
3
x
3 2 1
1
2
3
x
3 2 1
2
2
2
3
3
3
The lines may intersect.
The lines may be parallel.
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
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2
3
The lines may coincide.
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Example
Solve by graphing.
x
2
y
0
0
1
6
9
y =  3x  6
y
y = 2x  1
8
6
4
2
x
y
2  5
0 1
2
3
x
8
6
4
2
2
4
6
8
4
6
8
Continued
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Example (cont)
y
y = 2x  1
8
y =  3x  6
6
4
2
x
8
The lines
intersect at
(1, 3).
6
4
2
2
4
6
8
Check:
y = 3x  6
3 =  3(1)  6
3 = 3  6
3 = 3

4
6
8
The solution of the system is (–1, –3).
A system of equations that has one
solution is said to be consistent.
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
y = 2x  1
3 = 2(1)  1
3 = 2  1
3 = 3
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
Example
y
Solve by graphing.
3x  2y = 4
8
3x  2y = 4
 9x + 6y = 6
6
4
 9x + 6y = 6
2
x
8
The lines are parallel.
6
4
2
2
4
4
6
8
The lines do not intersect, so there is no solution.
A system of linear equations that has no
solution is called an inconsistent system.
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
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6
8
Example
y
The lines coincide.
Solve by graphing.
8
6
4x  6y = 8
 2x + 3y = 4
4
4x  6y = 8
2
x
8
6
 2x + 3y = 4
4
2
2
4
6
8
4
6
8
Every point on each graph coincides, thus there are an infinite
number of solutions.
A system of linear equations that has an infinite
number of solutions is called a dependent system.
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
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Example
Walter and Barbara need some plumbing repairs done at their
house. They called two companies for estimates of the work that
needs to be done. Roberts Plumbing and Heating charges $40
for a house call and then $35 per four for labor. Instant Plumbing
Repairs charges $70 for a house call and then $25 per hour for
labor.
a. Create a cost equation for each company, where y is the total
cost of plumbing repairs and x is the number of hours of labor.
Write the system of equations.
b. Graph the two equations using the values x = 0.3 and 6.
c. Determine from your graph how many hours of plumbing repairs
would be required for the two companies to charge the same.
d. Determine from your graph which company charges less if the
estimated amount of time is 4 hours.
Continued
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Example (cont)
a. Create a cost equation for each company, where y is the total
cost of plumbing repairs and x is the number of hours of labor.
Write the system of equations.
Total cost of
Cost of
=
plumbing
house call
cost per
+ hour
×
number of
labor hours
y
=
40
+
35
×
x
y
=
70
+
25
×
x
y = 40 + 35x
y = 70 + 25x
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
Continued
8
Example (cont)
b. Graph.
Roberts Plumbing and Heating
x
0
y
40
y
70
145
220
Y
240
220
200
y = 40 + 35x
3
145
6
250
Instant Plumbing Repairs
x
0
3
6
260
180
160
140
120
100
80
60
40
20
y = 70 + 25x
X
0
1
2
3
4
5
6
Continued
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
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Example (cont)
c. Determine from your graph how
many hours of plumbing repairs
would be required for the two
companies to charge the same.
260
Y
240
220
200
180
160
140
The lines intersect at (3, 145),
Thus the two companies will
charge the same if 3 hours of
plumbing repairs are required.
(3, 145)
120
100
80
60
40
20
X
0
1
2
3
4
5
6
Continued
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
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Example (cont)
d. Determine from your graph
which company charges less if
the estimated amount of time is
4 hours.
260
Y
240
220
200
180
160
We draw a dashed line at x = 4.
We see that the blue line is higher
than the red line. Thus, the cost
would be less if Walter and
Barbara use Instant Plumbing
Repairs for 4 hours of work.
140
120
100
80
60
40
20
X
0
1
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