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Lesson 7-1
Solving Systems by Graphing
Lesson Objectives
1 Solve systems by graphing
2 Analyze special types of systems
NAEP 2005 Strand: Algebra
Topics: Equations and Inequalities
Local Standards: ____________________________________
Vocabulary and Key Concepts.
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Numbers of Solutions of Systems of Linear Equations
different slopes
same slope
different y-intercepts
y
y
O
x
The lines
so there is
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solution.
O
x
The lines
so there are
solutions.
same slope
same y-intercept
y
O
x
The lines are
so there are
solutions.
A system of linear equations is
A solution of a system of linear equations is
No solution means
A system of equations has infinitely many solutions when
Daily Notetaking Guide
Algebra 1 Lesson 7-1
119
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Examples.
1 Solving a System of Equations Suppose you plan to start taking an
aerobics class. Nonmembers pay $4 per class, while members pay $10 a
month plus an additional $2 per class. What system of equations models the
cost of the class to members and nonmembers?
Define Let c
number of classes.
Let T(c) total cost of the classes.
plus
member
10
2
nonmember
0
4
Write
cost of classes
attended
2 Solve the system from the preceding Example.What does the
T(c) 2c 10
T(c) 4c
Total cost (dollars)
solution mean in terms of the original situation?
. The intercept on the
The slope is
.
vertical axis is
. The intercept on the
The slope is
.
vertical axis is
Graph the equations T(c) 2c 10 and T(c) 4c.
The lines intersect at
(
both will cost
). After
,
classes,
22
20
18
16
14
12
10
8
6
4
2
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membership
fee
cost
y
x
O 1 2 3 4 5 6 7 8 9
Number of classes
.
3 Systems With No Solution Solve by graphing. y 3x 2
y
y 3x 2
Graph both equations on the same coordinate plane.
y 3x 2
y 3x 2
The slope is
The slope is
The lines are
. The y-intercept is
x
.
. The y-intercept is
. There is
2
4 2
.
O
2
4
2
solution.
4 Systems With Infinitely Many Solutions Solve by graphing.
3x 4y 12
y
y 34x 3
Graph both equations on the same coordinate plane.
3x 4y 12
y 34x 3
The y-intercept is
The slope is
4
2
. The x-intercept is
. The y-intercept is
.
.
x
2 O
2
4
The graphs are the same line. The solutions are an
infinite number of ordered pairs (x, y), such that y 34x 3.
120
Algebra 1 Lesson 7-1
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
is
Relate
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Quick Check.
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1. You are testing two fertilizers on bamboo plants C and D. Plant C
is 5 cm tall and growing at a rate of 3 cm/day. Plant D is 11 cm tall
and growing at a rate of 4 cm/day. Write a system of equations that
models the height H(d) of each plant as a function of days d.
2. Two friends are walking around a quarter-mile track. One person
has completed six laps before the second one starts. The system
below models the distance d(t) in miles each walker covers as
a function of time t in hours.
d(t) 3t 1.5
d(t) 4t
a. Find the solution of the system by graphing.
d
Distance (mi)
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8
6
4
2
t
3
4
1
2
Second Walker’s Time (h)
0
b. What does the solution mean in terms of the original situation?
3. Without graphing, how can you tell if a system has no solution?
Give an example.
Daily Notetaking Guide
Algebra 1 Lesson 7-1
121