Transcript Chapter 8: Linear Equations as Models

Chapter 8: Linear
Equations as Models
Advanced Math
Section 8.1: Linear Growth and Decay
 Slope – Intercept Form of an Equation

y = mx +b ; m = slope ; b = y – intercept
 Slope


rise over run
change in y over the change in x
 Without graphing, find the slope and the vertical intercept of the
line modeled by each equation.
 Y = 7x + 4



Y = ½x + 17



Slope = 7
Y – intercept = 4
Slope = ½
y – intercept = 17
Try these on your own…

Y = 9 + 4x
 Slope = 4
 Y-intercept = 9

Y = 7/6x + 2
 Slope = 7/6
 Y-intercept = 2
Sample #1
 When you travel up from Earth’s surface, the air
temperature decreases by about 11 degrees for each
mile you rise about the ground. Suppose the
temperature of the air at ground level is 68 degrees.



Model the situation with a table of values.
Use your table to make a graph. Find the slope and the
vertical intercept of the line that contains the points.
Explain what they mean in terms of the situation.
Write an equation for the temperature of the air (t) as a
function of the height in miles above the ground (h).
Try this one on your own…
 Frankie has $500 in her savings account at
the beginning of the year. Each month she
saves $150.



Model the situation with a table of values and
a graph.
Find the slope and the vertical intercept of the
line. Explain what they mean in terms of the
situation.
Write an equation for the savings (s) as a
function of the time in months (t).
Linear Growth and Decay
 Linear Growth


Same Value
Repeatedly Added
Positive Slope
 Linear Decay


Same Value
Repeatedly Subtracted
Negative Slope
Sample #2
 Classify each situation as a linear growth, linear
decay, or neither. Explain your choice.



Eleanor Basave put $5 in a piggy bank for her
granddaughter when she was born. Every day after
that Eleanor put a quarter in the bank.
 Linear Growth
During each year, the shortest distance between Earth
and the sun is 94,000,000 miles and the greatest
distance is 94,500,000.
 Neither
A teacher on a remote Arctic island buys 144 cans of
fruit for the year. Each week the teacher eats 4 cans of
fruit.
 Linear Decay
Section 8.2: Linear Combinations
 A solution of an equation with two
variables is an ordered pair of numbers that
makes the equation true.
 All the points whose coordinates are solutions
of an equation form the graph of the
equation.
Standard Form of an Equation for a
Line
 The equation 8t + 12d = 48 is called a linear
equation because it is an equation of a line.
 Here is the standard form of a linear
equation…

ax + by = c
Sample #1
 Several times a week, Dynah Colwin runs
part of the way and walks part of the way on a
trail that is 4 miles long. Her running speed is
6 miles per hour and her walking speed is 4
miles per hour.

Write an equation for this situation.



Let r = the time in hours Dynah Colwin spends
running
Let w = the time in hours she spends walking
Rewrite the equation in slope-intercept form.
Try this one on your own…
 Each week, Will swims the backstroke part of
the way and the crawl part of the way for 30
laps of the swimming pool. His backstroke
speed is 3 laps per minute and his crawl
speed is 2 laps per minute.

Write an equation for this situation.



Let b = time it takes to swim the backstroke
Let c = time it takes to swim the crawl
Rewrite the equation in slope-intercept form.
Intercepts of a Line
 Vertical Intercept (y – intercept)

x=0
 Horizontal Intercept ( x – intercept)

y=0
Sample #2
 Find the intercepts
of the graph 2x – 5y
= 15. Use them to
graph the equation.
 Try this one on your
own…

Find the intercepts of
the graph of
y = -2x + 2. Use
them to graph the
equation.
Section 8.3: Horizontal and Vertical
Lines
 When you have a horizontal line, the slope is
zero.

The coefficient of x will be zero.
 When you have a vertical line, the slope is
undefined.

The coefficient of y will be zero.
Sample #1
 Find the slope of each line
and write an equation for
each line.
 A




The slope is 0.
The y-coordinate is
always 2.5.
Y = 2.5
B



The slope is
undefined.
The x-coordinate is
always -1.
X = -1
Try these on your own…
 Find the slope of each
line and write an
equation for each line.

A



Slope = 0
y = -7.6
B

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Slope = undefined
x = 1.2
Sample #2
 Write an equation for each line.

The point (-2,4) and (-2,0) are on the line.
 X-coordinates are both -2.
 Every point on the line has the same x coordinate, -2.
 X = -2

The slope is zero, and the point (2,3) is on the line.
 Slope is zero.
 Every point on the line has the same y coordinate, 3
 Y = 3
Try these on your own…
 Write an equation for each line.

The point (9,3) and (-1,3) are on the line.


y=3
The slope is undefined, and the point (-1,4) is
on the line.

x = -1
Section 8.4: Writing Equations for
Lines
 Estimate an equation
for the fitted line for
the Olympic data.

Step One: Find the
slope.

Step Two: Find the yintercept.

Step Three: Re-write
equation (y = mx + b)
filling in the slope and
y-intercept.
Try this one on your own…
 Estimate an
equation for the
fitted line for the
data. Interpret Week
0 to mean the
beginning of the first
week.



Slope = 4
y – intercept = 15
y = 4x + 15
Sample #2: Exact Equation from
Two Points
 Two points on a line are (4,3) and (-6,-2).
Write an equation for the line.

Step One: Find the slope.

Step Two: Find the y-intercept.

Step Three: Re-write equation (y = mx +b)
inputting the slope and y-intercept.
Try this one on your own…
 Two points on a line are (1, -2) and (5,4).
Write an equation for the line.

Step One: Find the slope.


Step Two: Find the y-intercept.

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slope = 3/2
y – intercept = -7/2
Step Three: Re-write equation.

y = 3/2x – 7/2
Sample #3: Equations from Other
Facts
 The slope of a line is 3/2. One point on the
line is (-4,1). Write an equation for the line.

Two Methods to Find the Y-Intercept

Substitution
 Use the Slope-Intercept Formula

Graph
Try this one on your own…
 The slope of a line is -1/2. One point on the
line is (3,4). Write an equation for the line.
Section 8.5: Graphing Systems of
Linear Equations
 MoviesPlus rents videos for $2.50 each and has
no membership fee. Videobusters rents videos
for $2 each but has a $10 membership fee.

Write and solve a system of equation to model this
situation. (Two Options – Graphing and Setting
Them Equal)



Let c = total cost of renting videos
Let n = number of videos rented
What advice would you give to someone trying to
decide which video store to use?
Try this one on your own…
 The service charge on a checking account at
Hometown Bank is $5 per month plus $0.15 for
each check written. The service charge at
Twentieth Century Bank is $0.25 per check.

Write and solve system of equations to model this
situation.



Let c = cost of the service charge
Let n = number of checks written
What advice would you give to someone trying to
decide which bank to use?
Sample #2: Systems without
Solutions
 Sandy and Rita are practicing for 100 m dash
competition. Sandy gives Rita a 10 m head
start.

Write and graph a system of equations to
model this situation.


Let x = time since the runner started
Let y = runner’s distance from starting line
Try this one on your own…
 James left the campground riding his bicycle
at 15 miles per hour. When he was 5 miles
away from camp, Lucie left the camp riding at
the same speed.

Write and graph a system of equations to
model this situation.


Let x = time Lucie started
Let y = their distance from the camp
Section 8.6: Graphing Linear
Inequalities
 Linear Inequalities on a Coordinate Plane

The graph of a linear inequality is a region on
a coordinate plane whose edge is a line.

The line is called a boundary line.
Sample #1: Graph each inequality.
yx
1
y
x4
2
Try these on your own…
y  x
1
y  x 1
2
Talk it Over…
 Tell whether the
boundary line for
each inequality is
solid or dashed.
Then tell whether
you would shade the
region above or
below the line.
y  9x  6
2
y x
3
y  4 x  11
1
y  x 
4
Sample #2
 Derek has saved $48 by baby-sitting. He
plans to use the money to buy tapes and
compact discs at a music store. Tapes cost
$8 and compact discs cost $12. The
inequality 8x  12y  48 represents the
amount Derek can spend at the store.

Graph the inequality 8x  12y  48
Try this one on your own…
 Graph the inequality 5y – 3x > 15.

Dashed or Solid Line?


Dashed
Shade Above or Below?

Below
Sample #3: Graphing Inequalities
in Standard Form
 Graph the inequality 8x – 12y < 48.

Use the intercepts to graph the boundary line.


Dashed or Solid?
Check test point.
Try this one on your own…
 Graph the inequality
5 y  3x  15