Slope of a Line

The slope of a line, denoted by m, is the rate of change between two points on a line and describes how much the line rises or falls as we move from left to right. It tells us how steep or flat the line is.

Formula

Mathematically,

Slope (m) = ${\displaystyle \frac{changeiny}{changeinx}}$

= ${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$

Here,

• (x₁, y₁) and (x₂, y₂) are two points on the line
• (y₂ – y₁) is the vertical change (rise)
• (x₂ – x₁) is the horizontal change (run)

The slope of a line can also be represented by 

m = tan θ = ${\displaystyle \frac{\mathrm{\Delta }x}{\mathrm{\Delta }y}}$

Rewriting the slope formula ${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$, we will get the point-slope form of the equation y – y₁ = m(x – x₁)

Derivation

Let us consider a straight line passing through the points A(x₁, y₁) and B(x₂, y₂)

Finding the Change in Coordinates

• ​The vertical change (rise) is Δy = y₂ – y₁
• The horizontal change (run) is Δx = x₂ – x₁ 

Finding the Slope

Now, from the right-angled triangle ABC, 

The slope is simply the tangent of the angle (θ) the line makes with the x-axis.

m = tan θ = ${\displaystyle \frac{oppositeside}{adjacentside}}$ 

= ${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$

Thus, the slope (m) = ${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$

Types of Slope

The slope of a line can be:

Positive (m > 0)

The line rises from left to right. An uphill road has a positive slope.

Negative (m < 0)

The line falls from left to right. A downhill road has a negative slope. 

Zero (m = 0)

It is a horizontal line. A flat road has zero slope.

Undefined

It is a vertical line. The slope of a wall or a tree trunk is undefined.

Finding the Slope 

From Two Points

If we are given two points on a line, say (x₁, y₁) and (x₂, y₂), we can calculate the slope by using the slope formula: 

m = ${\displaystyle \frac{changeiny}{changeinx}}$ = ${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$

Let us consider two points (3, 5) and (7, 11)

To find the slope:

Calculating the change in y-coordinates, 

y₂ – y₁ = 11 – 5 = 6

Calculating the change in x-coordinates,

x₂ – x₁ = 7 – 3 = 4

Now, finding the slope

m = ${\displaystyle \frac{changeiny}{changeinx}}$ = ${\displaystyle \frac{6}{4}}$ = ${\displaystyle \frac{3}{2}}$

Thus, the slope is: m = ${\displaystyle \frac{3}{2}}$

From an Equation

Sometimes, instead of two points, we are given the equation of a line. To determine the slope, we rewrite the equation in slope-intercept form:

y = mx + b …..(i)

For example, let us consider the equation: y = 4x – 7

By comparing it to the equation (i), we get

m = 4

Thus, the slope is m = 4.

From a Graph

We can also calculate the slope from a graph by choosing any two points on the line and counting the rise (vertical change) and run (horizontal change).

From the graph, we get the points (-5, 1) and (0, 6) 

Now, calculating the rise

Rise = y₂ – y₁ = 6 – 1 = 5

Counting the number of units the line moves left (run), we get

Run = x₂ – x₁ = 0 – (-5) = 5

Applying the slope formula:

m = ${\displaystyle \frac{rise}{run}}$ = ${\displaystyle \frac{5}{5}}$ = 1

Thus, the slope is m = 1

Solved Examples