Definition : The slope of a line passing
through two points 
and
is the
number m defined as
There are 4 different kinds of slope:
(1) positive
(2) negative
(3) zero
(4) infinity
Picture of lines through the point (a, b) with slopes
and −44
From the above picture of different lines of different slopes we can see that:
Equations of Horizontal and Vertical Lines
The equation of a horizontal line through the point (a, b) is y = b
The equation of a vertical line through the point (a, b) is y = a
Which of the fol lowing is a vertical line through the point (−3, 5).
(a) x = −3
(b) y = −3
(c) x = 5
(d) y = 5
Which of the following is a horizontal line through the point (−3, 5).
(a) x = −3
(b) y = −3
(c) x = 5
(d) y = 5
Homework 2.3 problems 27, 28
Definition : The point − slope form of a
line with slope m that goes through the point
is
Example : Let L denote the line that passes through the point (5, 10) and
through the
center of the circle (x−3)2 +(y +6)2 = 13. De termine the slope of L and the
equation of L.
The center of the circle is (3,−6)
The slope of L is 
The equation of L is y + 6 = 8(x − 3) or y − 10 = 8(x − 5).
Definition : The x − intercept of a line is the value at which the line crosses
the x-axis.
Definition : The y − intercept of a line is the value at which the line crosses
the y-axis.
To Find The x − intercept :
(1) Plug y = 0 into the equation of the line and solve for x .
To Find The y − intercept :
(1) Plug x = 0 into the equation of the line and solve for y.
Example : Find the equation of the line through the point (−1, 5) with
x-intercept 2.
Since the x- intercept is 2 the line crosses the x-axis at 2 so the point (2, 0)
is on the line.
So the equation of the line is
Homework 2.3 problems 13, 24c, 24d, 27, 28
Definition : The slope − intercept form of a line with slope m and y-intercept b
is
y = m x + b
Example Write the equation of a line with slope −1/2 and y-intercept −5
Example : What is the slope and y-intercept of a line
The slope of the line is 
and the y-intercept is b = −11
Homework 2.3 problems 13, 24c, 24d, 27, 28
Definition : The standard − form of a line is ax + by + c = 0 where a and b
cannot both
be 0.
Example : Find the area of the triangle that is created by the line 2x + 3y − 12
= 0 and
the x and y-axises.
2(0) + 3y − 12 = 0
3y = 12
y = 4 The y − intercept is (0, 4)
2x + 3(0) − 12 = 0
2x = 12
x = 6. The x − intercept is (6, 0)
So we can see the triangle has base 6 and height 4. So the area is
Example : Find the slope and y-intercept of the line 2x + 3y − 12 = 0.
3y = −2x + 12
The slope is
The y-intercept is b = 4
Homework 2.3 problems 13, 24c, 24d, 27, 28, 29
Definition : Given 2 lines, 
and
, the lines are parallel if
Definition : Given 2 lines,
and ,
the lines are perpendicular if
Example : Are the following 2 lines parallel:
and −6x + 8y + 16 = 0.
So 
and 
Since
the lines are parallel.
Example : Are the following 2 lines perpendicular:
and y = 2x − 7.
So 
and
So 
, so
. So no they are not perpendicular.
Example : Find the slope intercept form of the line that passes through the
point (2, 5) and
is perpendicular to the line 
The slope of our new line will be 
So
Homework 2.3 problems 13, 24c, 24d, 27, 28, 29, 31, 33, 35
Kinds Of Word Problems :
(1) Given a slope and point find something.
(2) Given two points find something.
(3) Given marginal cost, find something.
Definition : The marginal cost is the cost to produce one more item if n items
have already
been sold.
Definition Translation : If you have already produced 12 items, the marginal
cost is the
cost to produce the 13th item. In linear equations, it is always the slope.
Example : Suppose the cost C of producing widgets is a linear function of the
total number
of widgets produced w. If it costs 100 dollars to produce 20 widgets and 140
dollars to produce
25 widgets, what is the marginal cost of producing the 26th widget? Write an
equation in
slope-intercept form for the cost of producing widgets as a function of the
number of widgets
produced.
To find the cost of making the 26th widget we can find the cost of making 26
widgets and
subtract the cost of making 25 widgets. C = 8(26) − 60 = 208 − 60 = 148
148 − 140 = 8 So the marginal cost of making the 26th widget is 8 dollars.
Example : Suppose the cost C of producing a basketball is a linear function of
the number
of basketballs produced k. The marginal cost of producing a basketball is $5.50
and the cost
of producing 5 basketballs is $31. What is the cost of producing 12 basketballs?
C = 5.5k + b
31 = 5.5(5) + b
31 = 27.5 + b
b = 3.5
C = 5.5k + 3.5
C = 5.5(12) + 3.5
C = 66 + 3.5
C = 69.5
Example : A car is purchased for $20, 000. The car
de preciates by 25% in 4 years. If the
value of a newly purchaced car is a linear function of time, determine the
salvage value of
the car after 12 years.
.25 ยท 20, 000 = 5, 000
20, 000 − 5, 000 = 15, 000
Two points are (0, 20, 000) and (4, 15, 000).
Example : Fourteen years ago a house was worth $125, 000, now it is worth $550,
000.
As sume a linear relationship between the value of the home and time. What is the
house
going to be worth in 9 years, round your answer to the nearest penny.
Two points are (−14, 125, 000) and (0, 550, 000)
Example : The owner of a store can sell 12 carwashes per week at $10 a carwash
and can
sell 15 carwashes per week at $8 dollars per carwash. Assume that the sales S is
a linear
function of the price p of a carwash. Write an equation for S in terms of p. How
many
carwashes would she expect to sell at $6 per carwash?
Two points are (10, 12) and (8, 15).
18 carwashes.
