Tutorial on Slope of a Line

This is a tutorial on the slope of a line with examples and detailed solutions, and matched exercises also with solutions. If you wish to go first through a tutorial on the concept of the slope of a line Go here.

Example 1 : Find the slope of the line passing through the pairs of points and describe the line as rising, falling, horizontal or vertical.

  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).

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

  1. (3 , -1) , (3 , 5)
  2. (-1 , 0) , (3 , 7)
  3. (2 , 1) , (6 , 0)
  4. (-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 x1 and y1 be the x and y coordinates of point A. According to the definition of the slope
    ( y1 - 5 ) / (x1 - 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

  • We obtain an equation in y1
    ( y1 - 5 ) / -3 = -2

  • Solve for y1 to obtain
    y1 = 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?

  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 to 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.

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?

  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: Is it possible for two lines with negative slopes to be perpendicular?

Solution to Example 4:

No. 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.