Examples with Solutions

Example 1

1. (2 , 1) , (4 , 5)
2. (-1 , 0) , (3 , -5)
3. (2 , 1) , (-3 , 1)
4. (-1 , 2) , (-1 ,- 5)

Solution to Example 1

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.
   
2. 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.
   
3. 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).
   
4. 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).

Solution

1. (3 , -1) , (3 , 5)
2. (-1 , 0) , (3 , 7)
3. (2 , 1) , (6 , 0)
4. (-5 , 2) , (9 , 2)

Example 2

Solution to Example 2

1. Let x₁ and y₁ be the x and y coordinates of point A. According to the definition of the slope
   ( y₁ - 5 ) / (x₁ - 2) = -2
   
2. We need to solve this equation in order to find x₁ and y₁. This equation has two unknowns and therefore has an infinite number of pairs of solutions. We chose x₁ and then find y₁. if x₁ = -1, for example, the above equation becomes
   ( y₁ - 5 ) / (-1 - 2) = -2
   
3. We obtain an equation in y₁
   ( y₁ - 5 ) / -3 = -2
   
4. Solve for y₁ to obtain
   y₁ = 11
   
5. One possible answer is point A at
   (-1 , 11)
   
6. Check that the two points give a slope of -2
   (11 - 5 ) / (-1 - 2) = 6/-3 = -2

Solution

Example 3

1. L1: (1 , 2) , (3 , 1)
   L2: (0 , -1) , (2 , 0)
2. L1: (0 , 3) , (3 , 1)
   L2: (-1 , 4) , (-7 , -5)
3. L1: (2 , -1) , (5 , -7)
   L2: (0 , 0) , (-1 , 2)
4. L1: (1 , 0) , (2 , 0)
   L2: (5 , -5) , (-10 , -5)
5. L1: (-2 , 5) , (-2 , 7)
   L2: (5 , 1) , (5 , 13)

Solution Example 3

In what follows, m1 is the slope of line L1 and m2 is the slope of line L2.

1. 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.
   
2. 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.
   
3. m1 = ( -7 - (-1) ) / ( 5 - 2 ) = -6 / 3 = -2
   m2 = ( 2 - 0 ) / ( -1 - 0 ) = -2
   The two slopes are equal, the two lines are parallel.
   
4. 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
   
5. 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.

1. L1: (1 , 2) , (1 , 1)
   L2: (-4 , -1) , (-4 , 0)
2. L1: (2 , 3) , (3 , 1)
   L2: (1 , -2) , (7 , -5)
3. L1: (1 , -1) , (2 , -2)
   L2: (0 , 0) , (1 , 1)
4. L1: (1 , 9) , (-2 , 9)
   L2: (18 , -1) , (0 , -1)

Solution

Example 4

Solution to Example 4

No. If both slopes are negative, their product can never be equal to -1.

Solution

More References and links to topics on slopes.

• Slope of a Line - Interactive Java Applet.
  
• Easy to use calculator to find slope and equation of a line through two points.Two Points Calculator
  
• Another calculator to find slope, x and y intercepts given the equation of a line.Find Slope and Intercepts of a Line - Calculator