Lesson 1: Lines

Definition of a Line: Any equation that can be written in the form

Note: The equation above is called a linear equation because its graph is a line.
It is the general equation of the first degree in x and y.

Definition of Slope of a Line: If ( x1, y1 ) and ( x2 , y2 ) are any two distinct points on the
nonvertical line, then the slope of this line is given by
y − y1
m= 2
x2 − x1
Note:
a. In other words, the slope is the difference in the y values divided by the
difference in the x values. When using this definition do not worry about
which point should be the first point and which point should be the second
point. You can choose either to be the first and/or second and we will get
exactly the same value for the slope.
b. There is also a geometric definition of the slope of the line as well. You will
often hear the slope as being defined as follows,
rise
m=
run
Example 1.1: Determine the slope of each line that contains the two points below.
Then, draw a sketch of the graph of each line.
1. ( −2, −3 ) and ( 3,1)
2. ( −1,5 ) and ( 0, −2 )
Solution: To find the slope, all that we will need to do is use the slope formula. To
sketch the graph, we simply plot the two points and connect them with a line.

1− ( −3 ) 4
1. Slope: m = =
3 − ( −2 ) 5
4
Thus, the slope is . Below is a sketch of the line having this slope.
5
Graph:

Note: If the slope is positive, the line is increasing from left to right.

−2 − 5 −7
2. Slope: m = = = −7
0 − ( −1) 1
Hence, the slope is –7. Below is a sketch of the line having this slope.
Graph:
 Note: If the slope is negative, the line is decreasing as we move from left to
right.

Exercise 1.1: Determine the slope of each line that contains the two points below.
Then, draw a sketch of the graph of each line.
1. ( −3,2 ) and ( 5,2 )
2. ( 4,3 ) and ( 4, −2 )
Note:
a. The slope of a horizontal line is zero.
b. The slope of a vertical line is undefined.

Equations of a Line
1. Horizontal line: y = b, where b is the y intercept
2. Vertical line: x = a, where a is the x intercept
3. Point-slope form: y − y1 = m ( x − x1 )
4. Slope-intercept form: y = mx + b, where b is the y intercept
x y
5. Intercept form: + = 1, where a and b are the x and y intercepts
a b
6. Standard form: Ax + By = C
7. General form: Ax + By + C = 0
Note: For both standard and general forms, A, B, and C are constants and
A and B are not both zero.

Example 1.2: Write down the equation of the line that satisfies the following
conditions.
1. slope is 4 and passing through the point ( 2, −3 )
2. passing through the two points ( −2,4 ) and ( 3, −5 )

Solution:
1. Since the slope m = 4 and the point ( 2, −3 ) in which the line passes through
were given, we can use the point-slope form of an equation of the line. We
have
y − ( −3 ) = 4 ( x − 2 )
y + 3 = 4 ( x − 2)
y + 3 = 4x − 8
− 4 x + y + 11 = 0
−1( −4 x + y + 11) = −1( 0 )
4 x − y − 11 = 0
 Therefore, the equation of the line in general form is 4x − y − 11= 0 .
Note: For consistency in writing the final equation, we always transform the
resulting equation of the line in general form.

2. Let us first have a short analysis of the situation. We are given two points
( −2,4 ) and ( 3, −5 ) and we are asked to find the equation of the line passing
through these points. Since we have two points, we can obtain the slope
of the line. After getting the slope, we use either point to apply the point-
slope form of the linear equation. Let us now execute the plan.
First, we compute the slope. We have
−5 − 4 −9
m= =
3 − ( −2 ) 5
Then, we use the point-slope form of an equation of the line by considering
the computed slope and the given point ( −2,4 ) . We have
−9
y −4 =
5
( x − ( −2 ) )
−9
y −4 =
5
( x + 2)
 −9 
5 (y − 4) = 5  ( x + 2) 
 5 
5y − 20 = −9 ( x + 2 )
5y − 20 = −9 x − 18
9 x + 5y − 2 = 0
Therefore, the general form of the equation of the line is 9x + 5y − 2 = 0 .

Exercise 1.2: Write down the equation of the line that satisfies the following
conditions.
1. slope is –2 and passing through the point ( −3,5 )
2. passing through the points ( 4,6 ) and ( 0, −7 )
3. x intercept is –3 and the y intercept is 4

Enrichment Activity:
Theorem: Let l1 and l2 be two distinct nonvertical lines having slopes m1 and m2 ,
respectively. Then
i. l1 and l2 are parallel if and only if m1 = m2 .
ii. l1 and l2 are perpendicular if and only if mm 1 2 = −1
.
Problem: Determine if the line that passes through the points ( −2, −10 ) and ( 6, −1)
is parallel, perpendicular or neither to the line given by 7y − 9x = 15 .

Suggested Link: https://tutorial.math.lamar.edu/Classes/Alg/Lines.aspx

References:
Ayres, Jr., F. & Mendelson, E. (2009). Schaum’s Outlines: Calculus. (5th ed.). New
York: The McGraw-Hill Companies, Inc.
Krantz, S. (2003). Calculus Demystified. New York: The McGraw-Hill Companies,
Inc.
Leithold, L. (1989). College Algebra and Trigonometry. New York: Addison-
Wesley.
Leithold, L. (1990). The Calculus with Analytic Geometry (6th ed.). New York:
Harper & Row.