Linear Equations and Slope
Created by Laura Ralston
 http://www.youtube.com/watch?v=J_U93-
l5Z-w
Slope
 a useful measure of the “steepness” or “tilt”
of a line
 compares the vertical change (the rise) to
the horizontal change (the run) when
moving from one point to another along the
line
 typically represented by “m” because it is
the first letter of the French verb, monter
Formula and Graph
http://www.youtube.com/watch?v=xBdoD1RiNs
Four Possibilities of Slope

Positive Slope

• m>0
Line “rises” from left
to right
 Draw graph

Negative Slope
• m<0
Line “falls” from left to
right
 Draw graph

Four Possibilities of Slope

Zero Slope

• m=0
Line is horizontal
(constant)
 Draw graph

Undefined Slope
• m is undefined (0 in
denominator of ratio)
Line is vertical and is
NOT a function
 Do not say “NO slope”
 Draw graph

Using Slope to find the equation
of a line is IMPORTANT
Linear functions can take on many
forms
a) Point Slope Form
b) Slope Intercept Form
c) General Form
POINT-SLOPE FORM
Most useful symbolic
form
 Some explicit
information
 Not UNIQUE since any
point can be used, but
forms are equivalent
(graphs are identical)


Can use if
• slope and a point are
known
• or two points are
known
y = m(x - x1) + y1
Where m = slope of the line
and
(x1, y1) is any point on the line
Examples
 Straight forward:
Use the given conditions
to write the equation for each line. Write
final answer in slope intercept form
•
•
•
•
Slope =4, passing through (1, 3)
Slope = 53
, passing through (10, - 4)
Passing through (- 2, - 4) and (1, - 1)
Passing through (- 2, - 5) and (6, -5)
SLOPE INTERCEPT FORM
Most useful graphing
form
 Some explicit
information
 LIMITED in use
 UNIQUE to the graph


Can only be used if
slope and y-intercept
are known

To convert from pointslope to slope
intercept, apply the
distributive property.
y = mx + b
Where m = slope of the line
and
b = y-intercept
STANDARD FORM
Every line can be
expressed in this form
 No explicit
information


Ax + By = C
• where A, B, and C are
real numbers with A
not equal to 0
2 SPECIAL CASES

HORIZONTAL
• m=0
• y-intercept = b
• all points have the
same y-coordinate
• y = b or f(x) = b
– where b is any real
number

VERTICAL
•
•
•
•
m = undefined
no y-intercept
x-intercept = k
all points have same xcoordinate
• not a function
• x=k
– where k is any real
number
Examples
 Applications
• A business purchases a piece of equipment for
$30,000. After 15 years, the equipment will
have to be replaced. Its value at that time is
expected to be $1,500. Write a linear equation
giving the value, y, of the equipment in terms of
x, the number of years after it is purchased.
What is the value of the equipment 5 years
after it is purchased?
Examples
 Applications:
• In 1999, there were 4076 JC Penney stores and
in 2003, there were 1078 JC Penney stores.
Write a linear equation that gives the number
of stores in terms of the year. Let t = 9
represent 1999. Predict the number of stores
for the year 2008. Is your answer reasonable?
Explain.
Examples
A discount outlet is offering a 15% discount on all
items. Write a linear equation giving the sale
price S for an item with a list price x.
 Dell Computers Inc pays its mircochip assembly
line workers $11.50 per hour. In addition workers
receive a piecework rate of $0.75 per unit. Write a
linear equation for the hourly wage W in terms of
the number of units x produced per hour

SPECIAL LINEAR RELATIONSHIPS
 PARALLEL :
Two or more lines that run side
by side
• never intersecting
• always same distance apart
• each line has the same slope m1 = m2
 PERPENDICULAR : Two lines that intersect to
form 4 right angles
• Product of the slopes is equal to -1
m1m2 = -1
Examples
 Passing through (-8, -10) and parallel to the
line, y = - 4x + 3
 Passing through (- 4, 2) and perpendicular
to the line, y = ½x + 7
 Passing through (- 2, 2) and parallel to the
line, 2x – 3y – 7 =0
 Passing through (5, - 9) and perpendicular
to the line, x + 7y – 12 = 0