Basic ingredients of linear modeling-BBA-Mathematics.pdf

BBAG1353
WEEK NO. 3
Basic Ingredients of Mathematical Models

Linear Function
• A linear function is one having two
variables such that one variable depends
on the other.
For example:
Y = ax + b
In this function, y depends on the variable x

Representing a
linear equation
Y=mx + c

What is Slope of a Line (m)?
Slope basically describes the
steepness of a line
Slope is simply the change in the vertical
distance over the change in the horizontal
distance

Measurement of Slope
Slope is simply the change in the
vertical distance over the change in
the horizontal distance
1
2
1
2
x
x
y
y
x
y
run
rise
m
slope









1
2
1
2
x
x
y
y
m



The formula above is the one which we
will use to find the slope of specific lines
In order to use that formula we need to
know, or be able to find 2 points on the
line

If a line is in the form Ax + By = C,
we can use the following formula to
find the slope:
B
A
m 


Examples
   
 
3
1
6
2
1
5
4
6
6
,
5
,
4
,
1







m
m
3
2
5
3
2




m
y
x

Horizontal lines have a slope of zero
while vertical lines have no slope
Horizontal
y=
Vertical
x=
m = 0
m = no
slope

• If a line goes up from
left to right, then the
slope has to be positive
Conversely, if a line goes down from left
to right, then the slope has to be negative

Estimating the slope and
intercept through given
equation

To find the slope and y-intercept of
an equation, write the equation in
slope-intercept form: y = mx + b.
Find the slope and y-intercept.
1) y = 3x – 7
y = mx + b
m = 3, b = -7

2) y = x
y = mx + b
y = x + 0
3) y = 5
y = mx + b
y = 0x + 5
2
3
m =
b = 0
2
3
2
3
m = 0
b = 5

-3 -3 -3
4) 5x - 3y = 6
Write it in slope-intercept form. (y = mx + b)
5x – 3y = 6
-3y = -5x + 6
y = x - 2
5
3
m =
b = -2
5
3

Write it in slope-intercept form. (y = mx + b)
2y + 2 = 4x
2y = 4x - 2
y = 2x - 1
5) 2y + 2 = 4x
2 2 2
m = 2
b = -1

Find the slope and y-intercept of
y = -2x + 4
1. m = 2; b = 4
2. m = 4; b = 2
3. m = -2; b = 4
4. m = 4; b = -2

Interpretation of slope in an equation
• Slope of the equation tells about the change in Dependent
variable due to unit change in independent variable.
• For example:
In a linear equation : y = 2x + 5, the slope is 2 which tells that
the value of y (DV) will change by 2 units due to unit change
in value of x (IV).

Interpretation of intercept
• Intercept in an equation is a point where the line cuts at x-
axis or y-axis
• y-intercept is the point where the line cuts at y-axis.
• x-intercept is the point where the line cuts at x-axis.
• The intercept shows the value of DV when value of IV is
zero.

Writing linear
equations
Writing Linear Equations
In Slope-Intercept Form
y = mx + b

If you are given:
The slope and y-intercept
¡ Finding the equation of
the line in y= mx + b
form. Given: slope
and y-intercept. Just
substitute the “m” with
the slope value and the
“b” with the y-intercept
value.
¡ Slope = ½ and
¡ y-intercept = -3
y= mx + b
½
-3
y= ½x – 3

If you are given: A Graph
Find the:
¡ y – intercept = b = the point
where the line crosses the y
axis.
¡ Slope = = m =
run
rise
s
x'
in
change
s
y'
in
change
¡ y – intercept = b = -3
¡ Slope = = m = ½
y= mx + b
2
over
1
up ½
-3
y= ½x – 3

If you are given:
The slope and a point
¡ Given: slope (m)
and a point (x,y).
To write equations
given the slope and
a point using Point-
Slope Form.
¡ Slope =½ and point (4,-1)
½ 4-1
y= ½x – 3
Point-Slope Form
1 1
y y m(x x )
  
1 1
y y m(x x )
  
   
4
2
1
1 


 x
y
 
4
2
1
1 

 x
y
2
2
1
1 

 x
y
-1 -1

If you are given:
Two points
¡ Finding the equation
of the line in y= mx +
b form. Given: Two
points. First find the
slope (m) and then
substitute one of the
points x and y values
into Point-Slope Form.
1 1
y y m(x x )
  
Point-Slope Form
Point (-2, -4) & Point (2, -2)
Find the:
n Slope = = m =
s
x'
in
change
s
y'
in
change
run
rise
 
 






2
2
4
2




2
2
4
2
2
1
4
2

¡ Slope =½ and point (2, -2)
1 1
y y m(x x )
  
½ -2
2
   
2
2
1
2 


 x
y
1
2
1
2 

 x
y
-2 -2
y= ½x – 3

Write the equation of a line that has
a y-intercept of -3 and a slope of -4.
1. y = -3x – 4
2. y = -4x – 3
3. y = -3x + 4
4. y = -4x + 3

Write an equation of the line that goes
through the points (0, 1) and (1, 4).
1. y = 3x + 4
2. y = 3x + 1
3. y = -3x + 4
4. y = -3x + 1

Practice Questions
Write an equation to model each situation.
1. You rent a bicycle for $20 plus $2 per hour.
2. An auto repair shop charges $50 plus $25 per hour.
3. A candle is 6 inches tall and burns at a rate of inch per
hour.
4. In 1995, Orlando, Florida, was about 175,000. At that time,
the population was growing at a rate of about 2000 per year.
Write an equation to find Orlando’s population for any year.
Also predict what Orlando’s population will be in 2010.

5. Couples are marrying later. The median age of men who
tied the knot for the first time in 1970 was 23.2. In 1998, the
median age was 26.7. Write an equation to predict the median
age that men marry M for any year t. Also predict the median
age of men who marry for the first time in 2005.

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