Slope of a Line

Slopes of lines are studied through definitions, examples with detailed solutions and an interactive app.

Definition of the Slope of a Line

The slope of a straight line is a measure of how steep a line is and in general it is a measure of change. It is an extremely important concept in mathematics.
The three lines shown below have different properties: line \( L_1 \) is steeper than line \( L_2 \). Lines \( L_1 \)and \( L_2 \) rise but line \( L_3 \) falls. A parameter is needed to measure all these properties of a line quantitatively.


three lines with different slopes
Let us consider the two points \( P_1(x_1,y_1) \) and \( P_2(x_2,y_2) \) on the line shown below. As we move from point \( P_1 \) to point \( P_2 \); to a change \( \Delta y = y_2 - y_1\) in the vertical direction corresponds a change \( \Delta x = x_2 - x_1\) in the horizontal direction.
The slope \( m \) of a line is defined as the ratio of the change along the y axis and the change of the x axis
\[ m = \dfrac{\Delta y}{\Delta x} = \dfrac {y_2 - y_1}{x_2 - x_1} \]
definition of a slope

Properties of the Slope of a Line

1) The slope of a line is constant and any two points on the line may be used to find its value.
2) A line with positive slope rises from left to right.
3) A line with negative slope falls from left to right.
4) The slope of a horizontal line is equal to 0.
5) The slope of a vertical line in undefined,
6) Since the slope is a ratio, it is also called rate of change.
7) Two lines with slopes \( m_1 \) and \(m_2 \) are parallel (they do not have a point of intersection) if and only if \( m_1 = m_2 \)
8) Two lines with slopes \( m_1 \) and \(m_2 \) are perpendicular (they have a point of intersection and make a right angle) if and only if \( m_1 \times m_2 = - 1 \)
9) Although gradient and slope are sometimes used interchangeably, the gradient is a vector and the slope is a scalar. The magnitude of the gradient of a line is equal to the absolute value of the slope of this line.



Example 1 Find slopes
Find the slope of the lines \( L_1, L_2, L_3, L_4 \) through the given pair of points and describe the lines as rising, falling, horizontal or vertical and plot the given pair of points and the corresponding line in the same system of coordinates.

  1. \( L_1: (2 , 3) , (4 , 9) \)
  2. \( L_2: (-1 , 0) , (6 , -8) \)
  3. \( L_3: (-5 , 1) , (-5 , 3) \)
  4. \( L_4: (7 , -3) , (-5 , -3) \)

Solution to Example 1
  1. \( L_1: m_1 = \dfrac{9 - 3}{4 - 2} = 3\) , the line has a positive slope and therefore rises.

  2. \( L_2: m_2 = \dfrac{-8 - 0}{6 - (-1)} = -\dfrac{8}{7} \) , the line has a negative slope and therefore falls.

  3. \( L_3: m_3 = \dfrac{3 - 1}{- 5 - (-5) } = \dfrac{2}{0} = \text{undefined} \) , the slope is undefined and therefore the line is vertical.

  4. \( L_4: m_3 = \dfrac{-3 - (-3)}{-5 - 7 } = \dfrac{0}{-12} = 0 \) , the slope is equal to 0 and therefore the line is horizontal.

    The graphs of the four lines and the pairs of points are shown below where all the above answers are confirmed.

    Graph of lines with different slopes



Example 2 slopes of parallel and perpendicular lines
Find the slope of the lines \( L_1, L_2, L_3\) through the given pair of points and describe the lines as parallel or perpendicular. Graph the three lines in the same system of coordinates.

  1. \( L_1: (0 , 0) , (2 , 4) \)
  2. \( L_2: (0 , 0) , ( - 2 , 1) \)
  3. \( L_3: (1 , 1) , (3 , 5) \)

Solution to Example 2
  1. \( L_1: m_1 = \dfrac{4 - 0}{2 - 0} = 2 \)

  2. \( L_2: m_2 = \dfrac{1 - 0}{-2 - 0} = -\dfrac{1}{2} \)

  3. \( L_3: m_3 = \dfrac{5 - 1}{3 - 1} = 2 \)

    Since \(m_1 = m_3 \) , lines \( L_1 \) and \( L_3 \) are parallel.
    Since \(m_1 \times m_2 = 2 \times (-1/2) = - 1 \) and \(m_3 \times m_2 = 2 \times (-1/2) = - 1 \), \( L_1 \) and \( L_2 \) are perpendicular and \( L_3 \) and \( L_2 \) are also perpendicular.
    The graphs of the three lines are shown below.

    Graph of parallel and perpendicular lines



Example 3
Two lines \( L_1 \) and \( L_2 \) are defined by the pair of points: \( L_1: (-1,-1), (2,3) \) and \( L_2: (1,2) , (4,7) \)
a) Find the slope of the two lines. Which of the two lines rises faster?
b) Graph the two lines and confirm graphically your answer to part a).

Solution to Example 3
a) \( L_1: m_1 = \dfrac{3 - (-1)}{2- (-1)} = \dfrac{4}{3} \)

\( L_2: m_2 = \dfrac{7 - 2}{4 - 1} = \dfrac{5}{3} \)

Both lines have positive slopes and therefore rise. But because the slope \( m_2 = \dfrac{5}{3} \) is larger than the slope \( m_1 = \dfrac{4}{3} \), line \( L_2 \) rises faster than line \( L_1 \).
b) The graphs of the two lines are shown below and the answer to part a) is confirmed.

Graph of two lines to compare their slopes

Interactive Tutorial to Explore the Slope of a Line

An app that can be used to explore the slope of a line interactively using the mousse is shown below. Follow the instructions below and explore the slope.
Given two points \( A(x_A , y_A) \) and \( B(x_B , y_B) \), the slope m of the line through the two points is given by the formula (see formula above):

\( m = \dfrac{y_B - y_A}{x_B - x_A} \)

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  1. Use the mousse, click press and drag points A and B slowly, determine their coordinates and use the formula of the slope of a line to calculate the slope of the line through the two points.
  2. Keep A in a fixed position and change the position of point B so that the line through AB is parallel to the x axis. Calculate the slope.
  3. Keep A in a fixed position and change the position of point B so that the line through AB is parallel to the y axis. Calculate the slope.
  4. Keep A in a fixed position and change the position of point B so that the line rises. Calculate the slope. Is the slope positive, negative or equal to zero?
  5. Keep A in a fixed position and change the position of point B so that the line falls. Calculate the slope. Is the slope positive, negative or equal to zero?


More References and Links

Solve Slope Problems
Slopes of Parallel Lines Questions
Easy to use calculator to find
slope and equation of a line through two points.Two Points Calculator