Examples with SolutionsExample 1
Find the slope of the line
passing through the pairs of points and describe the line as rising, falling,
horizontal or vertical.
 (2 , 1) , (4 , 5)
 (1 , 0) , (3 , 5)
 (2 , 1) , (3 , 1)
 (1 , 2) , (1 , 5)
Solution to Example 1

The slope of the line is given by
m = ( 5  1 ) / (4  2) = 4 / 2 = 2
Since the slope is positive, the line rises as x increases.

The slope of the line is given by
m = ( 5  0 ) / ( 3  (1) ) = 5 / 4
Since the slope is negative, the line falls as x increases.

We first find the slope of the line
m = ( 1  1 ) / ( 3  2 ) = 0
Since the slope is equal to zero, the line is horizontal (parallel to the x axis).

The slope of the line is given by
m = ( 5  2 ) / ( 1  (1) )
Since ( 1  (1) ) = 0 and the division by 0 is not defined, the slope of the line is undefined and the line is vertical. (parallel to the y axis).
Matched Exercise to Example 1
Find the slope of the line passing through the pairs of points and describe the line as rising, falling, horizontal or vertical.
Solution
 (3 , 1) , (3 , 5)
 (1 , 0) , (3 , 7)
 (2 , 1) , (6 , 0)
 (5 , 2) , (9 , 2)
Example 2
A line has a slope of 2 and passes through the point (2 , 5). Find another point A through which the line passes. (many possible answers)
Solution to Example 2

Let x_{1} and y_{1} be the x and y coordinates of point A. According to the definition of the slope
( y_{1}  5 ) / (x_{1}  2) = 2

We need to solve this equation in order to find x_{1} and y_{1}. This equation has two unknowns and therefore has an infinite number of pairs of solutions. We
chose x_{1} and then find y_{1}. if x_{1} = 1, for
example, the above equation becomes
( y_{1}  5 ) / (1  2) = 2

We obtain an equation in y_{1}
( y_{1}  5 ) / 3 = 2

Solve for y_{1} to obtain
y_{1} = 11

One possible answer is point A at
(1 , 11)

Check that the two points give a slope of 2
(11  5 ) / (1  2) = 6/3 = 2
Matched Exercise to Example 2
A line has a slope of 5 and passes through the point (1 , 4). Find another point A through which
the line passes. (many possible answers)Solution
Example 3
Are the lines L1 and L2
passing through the given pairs of points parallel, perpendicular or
neither parallel nor perpendicular?
 L1: (1 , 2) , (3 , 1)
L2: (0 , 1) , (2 , 0)
 L1: (0 , 3) , (3 , 1)
L2: (1 , 4) , (7 , 5)
 L1: (2 , 1) , (5 , 7)
L2: (0 , 0) , (1 , 2)
 L1: (1 , 0) , (2 , 0)
L2: (5 , 5) , (10 , 5)
 L1: (2 , 5) , (2 , 7)
L2: (5 , 1) , (5 , 13)
Solution Example 3In what follows, m1 is the slope of line L1 and m2
is the slope of line L2.

Find the slope m1 of line L1 and the slope m2 of line L1
m1 = ( 1  2 ) / ( 3  1 ) = 1 / 2
m2 = ( 0  (1) ) / ( 2  0 ) = 1/2
The two slopes m1 and m2 are not equal and
their products is not equal to 1. Hence the two lines are neither parallel nor
perpendicular.

m1 = ( 1  3 ) / ( 3  0 ) = 2 / 3
m2 = ( 5  4 ) / ( 7  (1) ) = 9 / 6 = 3/2
The product of the two slopes m1*m2 = (2
/ 3)(3 / 2) = 1, the two lines are perpendicular.

m1 = ( 7  (1) ) / ( 5  2 ) = 6 / 3 = 2
m2 = ( 2  0 ) / ( 1  0 ) = 2
The two slopes are equal, the two lines
are parallel.

m1 = ( 0  0 ) / ( 2  1 ) = 0 / 1 = 0
m2 = ( 5  (5) ) / ( 10  5 ) = 0 / 15 = 0
The two slopes are equal , the two lines
are parallel. Also the two lines are horizontal

m1 = ( 7  5 ) / ( 2  (2) )
m2 = ( 13  1 ) / ( 5  5 )
The two slopes are both undefined since the denominators in both m1 and m2 are equal to zero. The two lines are vertical lines and therefore parallel.
Matched Exercise to Example 3
Are the lines L1 and L2 passing through the given pairs of points parallel, perpendicular or neither parallel nor perpendicular?
 L1: (1 , 2) , (1 , 1)
L2: (4 , 1) , (4 , 0)
 L1: (2 , 3) , (3 , 1)
L2: (1 , 2) , (7 , 5)
 L1: (1 , 1) , (2 , 2)
L2: (0 , 0) , (1 , 1)
 L1: (1 , 9) , (2 , 9)
L2: (18 , 1) , (0 , 1)
Solution Example 4
Is it possible for two
lines with negative slopes to be perpendicular?
Solution to Example 4No. If both slopes are negative, their
product can never be equal to 1. Matched Exercise to Example 4
Is it possible for two lines with positive slopes to be perpendicular?Solution More References and links to topics on slopes. 