Chabot Mathematics
§2.4b Lines
by m & b
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Review §
2.4
MTH 55
 Any QUESTIONS About
• §’s2.4 → Intercepts, Slopes
 Any QUESTIONS About HomeWork
• §’s2.4 → HW-06
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
The Slope-Intercept Equation
 The equation y = mx + b is called
the slope-intercept equation.
 The equation represents a line of
slope m with y-intercept (0, b)
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example Find m & b
 Find the slope and the y-intercept of
each line whose equation is given by
3
a) y  x  2 b) 3x  y  7
c) 4 x  5 y  10
8
 Solution-a)
Slope is
3/8
Chabot College Mathematics
4
3
y  x2
8
InterCept
is (0,−2)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Find m & b cont.1
 Find the slope and the y-intercept of
each line whose equation is given by
3
a) y  x  2 b) 3x  y  7
c) 4 x  5 y  10
8
 Solution-b) We first solve for y to find an
equivalent form of y = mx + b.
y  3x  7
 Slope m = −3
 Intercept b = 7
• Or (0,7)
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Find m & b cont.2
 Find the slope and the y-intercept of
each line whose equation is given by
3
a) y  x  2 b) 3x  y  7
c) 4 x  5 y  10
8
 Solution c) rewrite the equation in the
4
form y = mx + b.
y  x2
5
4 x  5 y  10
4 x  10  5 y
1
5 y  4 x  10
5
Chabot College Mathematics
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 Slope, m = 4/5
(80%)
 Intercept b = −2
• Or (0,−2) Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Find Line from m & b
 A line has slope −3/7 and y-intercept
(0, 8). Find an equation for the line.
 We use the slope-intercept equation,
substituting −3/7 for m and 8 for b:
3
y  mx  b   x  8
7
 Then in y = mx + b
Form
Chabot College Mathematics
7
3
y   x 8
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Graph y = (4/3)x – 2
 SOLUTION: The
slope is 4/3 and the
y-intercept is (0, −2)
right 3
up 4 units
 We plot (0, −2) then
move up 4 units and
to the right 3 units.
Then Draw Line
 We could also move
down 4 units and to
the left 3 units. Then
draw the line.
Chabot College Mathematics
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(0, 2)
down 4
(3, 6)
left 3
4
y  x2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
(3, 2)
Example  Graph 3x + 4y = 12
 SOLUTION: Rewrite the equation in
slope-intercept form
3x  4 y  12
4 y  3x  12
4 y  3 x  12
4
3
y   x3
4
Chabot College Mathematics
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 Thus
• m = −3/4
– Rise = −3
– Run = 4
• b=3
– or (0, 3)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Graph 3x + 4y = 12
 SOLUTION: The
slope is −3/4 & the
y-intercept is (0, 3).
 We plot (0, 3), then
move down 3 units
and to the right 4
units to Plot Line
 An alternate
approach would be
to move up 3 units
and to the left 4 units
Chabot College Mathematics
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left 4
(4, 6)
up 3
(0, 3)
down 3
right 4
3x  4 y  12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
(4, 0)
Parallel Lines by Slope-Intercept
 Slope-intercept form allows us to quickly
determine the slope of a line by simply
inspecting, or looking at, its equation.
 This can be especially helpful when
attempting to decide whether two
lines are parallel
 These Lines All
Have the SAME
Slope
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Parallel Lines
 Determine whether the graphs of these
two Equations are Parallel (||):.
3
y  x3
2
3x  2 y  5
 SOLUTION: Remember that parallel
lines extend indefinitely without
intersecting. Thus, two lines with the
SAME SLOPE but different
y-intercepts are PARALLEL
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Parallel Lines cont.
 The line (3/2)x+3 has slope 3/2
and y-intercept 3
3x  5  2 y
 We need to rewrite

3x−2y = −5 in
3
5
y  x
slope-intercept form:
2
2
 slope is 3/2 and the y-intercept is 5/2.
 Both lines have slope 3/2 and
different y-intercepts; thus the
graphs ARE parallel.
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Parallel Line Example
14
3
y  x3
2
12
Parallel
Lines
10
8
y
3
5
y  x
2
2
6
4
2
0
0
1
file = M65_§3-5_Graphs_0607.xls
Chabot College Mathematics
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2
3
4
5
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
6
Perpendicular Lines
 In the coordinate
plane, two lines are
perpendicular if the
product of their
slopes (m) is −1.
 In This Example
1
mup 
mdn  2
2
 Then
1
mup mdn     2  1
2
 
Chabot College Mathematics
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
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
RATE Defined
 A RATE is a ratio that indicates
how two quantities change with
respect to each other
 Some Examples
• Miles per Gallon (mpg) → Fuel Efficiency
• $ per Pound → Food Cost
• kg per Cubic-Meter (kg/m3) → Density
• $ per Hour → Wage Rate
• Yards per Catch → Football Receiving
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Rates on Rental Car

On March 4, Nichole rented a mini-van with
a full tank of gas and 10,324 mi on the
odometer. On March 9, she returned the
mini-van with 10,609 mi on the odometer. If
the rental agency charged Nichole $126 for
the rental and needed 15 gal of gas to fill up
the gas tank, find the following rates:
a) The car’s average rate of gas consumption,
in miles per gallon.
b) The average cost of the rental, in dollars per day.
c) The car’s avg. rate of travel, in miles per day.
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Rates on Rental Car
 Solution a) Fuel Use Rate
• Change in Fuel = 15 gal
• Change in Distance = (10 609 − 10 324) mi
 The RATE of CHANGE
Dist Change 10609mi  10324mi 285mi
mi


 19
Fuel Change
15 gal
15 gal
gal
 The RATE of CHANGE is 19 mpg
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Rates on Rental Car cont.1
 Solution b) $ per Day
• Change in Money = $126
• Change in Time = 09Mar − 04Mar = 5 Days
 The RATE of CHANGE
Money Chg $126
$

 25.20
Time Change 5day
Day
 The RATE of CHANGE is
$25 & 20¢ Per Day
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Rates on Rental Car cont.2
 Solution c) Miles per Day
• Change in Distance = (10 609 − 10 324) mi
• Change in Time = 09Mar − 04Mar = 5 Days
 The RATE of CHANGE
Dist Chg 10609mi  10324mi 285mi
mi


 57
Time Chg
5day
5day
day
 The RATE of CHANGE is
57 miles Per Day
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Rate of Change
 Alonzo’s Hair Salon
has a graph
displaying data from a
recent day of work.
a) What rate can be
determined from the
graph?
1
2
3
4
b) What is that rate?
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
5
Example  Rate of Change
 The Quantity
Changes
•
•

Change In HairCuts
= 10 − 2 = 8
Change in Time =
5pm−1pm = 4 hours
Thus the
PRODUCTION Rate
1
2
3
4
8 HairCuts
HairCuts
Production Rate 
2
4 hours
hour
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
5
Example  Using Rates
 Madhuri has a home healthcare
business, specializing in
physical therapy.
 Her weekly income is directly
proportional to the number of patients
she sees each week.
 If she gets paid $33 per session,
what will be her income if she sees
16 patients a week?
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Using Rates
 Translating: LET
• i be her weekly income
• n be the number of patients she sees
in a week
• p be the amount she gets paid per session;
i.e; p is the service RATE.
 In Equation Form i = p•n
• If n = 16 Patients per Week
$33

 16 Patient  Session  $528
i  pn  


1 - Week
 Patient  Session 
 Week
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Modeling Data by y = mx + b
 Curve Fitting/Modeling
• In general, we try to find a function that fits,
as well as possible, observations (data),
theoretical reasoning, and common sense.
 EXAMPLE
• Model the data given in the plot on foreign
travel on the next slide with two different
linear functions. Then with each function,
predict the number of U.S. travelers to
foreign countries in yr 11. Of the two
models, which appears to be the better fit?
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Model by mx + b
 Given Data in Plot
 For Model-I draw a
“Good” Line thru the
Data in the Plot
 Find Slope using
Two points on the
Line (yrs 1 & 5)
y2  y1 5.75  5.08
mI 

x2  x1
5 1
mI  0.1675
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Model by mx + b
 Examine Model-I
Line to Estimate
Intercept
bI  4.91 by EyeBall 
 The Model-I Linear
Equation
yI  mI x  bI
yI  0.1675 x  4.91
Chabot College Mathematics
27
 Travelers at Yr-11
yI 11  0.167511  4.91
yI 11  6.7525
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Model by mx + b
 Given Data in Plot
 For Model-II draw a
“Good” Line thru the
Data in the Plot
 Find Slope using
Two points on the
Line (yrs 0 & 6)
y2  y1 6.08  4.65
mII 

x2  x1
60
mII  0.2383
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Model by mx + b
 Examine Model-II
Line to Estimate
Intercept
bII  4.65 by Data 
 The Model-II Linear
Equation
yII  mII x  bII
yII  0.2383x  4.65
Chabot College Mathematics
29
 Travelers at Yr-11
yII 11  0.238311  4.65
yII 11  7.2713
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Example  Compare Models
 Model-I predicts about
6.76 million U.S. foreign
travelers in Yr-11 while
Model-II predicts about
7.27 million.
 It appears from the
graphs that Model-II fits
the data more closely,
thus we would choose
Model-II over Model-I.
• A Close Call
Chabot College Mathematics
30
yI  0.1675x  4.91
yII  0.2383x  4.65
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
WhiteBoard Work
 Problems From
§2.4 Exercise Set
• PPT → 78, 80
• 34, 44, 74

HipHop &
HomePrices
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
P2.4-78  Rap/HipHop
 Find Average Rate-ofChange for HipHop
Sales over 1997-2002
 Connect ’97 & ’02
Dots to Reveal Avg Rt
 Read Graph to Find
(x1, y1) and (x2, y2)
• (x1, y1) = (1997, 10.1%)
• (x2, y2) = (2002, 13.8%)
 Recall That the Rate
is also the Slope
Chabot College Mathematics
32
y2  y1 13.8%  10.1%
Rt 

x2  x1 2002  1997 yrs
Rt  0.74 % yr
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
P2.4-80  Home Sale $-Price
 From Data Produce
Model: S(x) = mx + b
 Use Labeled End-Pts
to find Slope, m
y2  y1
m
x2  x1
$149 900  $128 400
m
4yrs - 0yrs
m  $5375 yr
 b is pt at y = 0 →
b  yx  0  $128 400
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
P2.4-80  Home Sale $-Price
 Thus the Model:
S(x) = mx + b
 $5375 
S
x  $128 400

 yr 
 Use Model to Find
S(2010)
x  2010  1998  12 yr
 $5375 
12 yr   $128 400
S 2010  

 yr 
S 2010  $192 900
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
All Done for Today
Slope of a
CURVE
by Calculus
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt