What Is the Gradient (Slope) of a Line?

Before we explore the different forms of linear equations, we need to understand the most important property of a straight line: its gradient (also called slope).

Gradient (Slope)

The gradient of a line measures how steep it is. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It is usually represented by the letter $m$ .

The formula for gradient is:

$m = \frac{rise}{run} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

where $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are any two points on the line.

Analogy

Think of gradient like the steepness of a hill. A gentle slope has a small gradient (like $m = 0.5$ ), while a cliff-like slope has a large gradient (like $m = 10$ ). Walking downhill means the gradient is negative, and a perfectly flat path has a gradient of zero.

Key facts about gradient:

A positive gradient means the line goes upward from left to right ↗
A negative gradient means the line goes downward from left to right ↘
A gradient of zero means the line is horizontal →
An undefined gradient means the line is vertical ↑

Calculating Gradient from Two Points

Let's practise finding the gradient using the formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ .

Example

Find the gradient of the line passing through $(2, 3)$ and $(6, 11)$ .

Step 1: Label the points.

$(x_{1}, y_{1}) = (2, 3)$
$(x_{2}, y_{2}) = (6, 11)$

Step 2: Substitute into the formula.
$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$

The gradient is $m = 2$ . This means for every 1 unit you move to the right, the line goes up 2 units.

Example

Find the gradient of the line passing through $(- 1, 5)$ and $(3, - 3)$ .

Step 1: Label the points.

$(x_{1}, y_{1}) = (- 1, 5)$
$(x_{2}, y_{2}) = (3, - 3)$

Step 2: Substitute into the formula.
$m = \frac{- 3 - 5}{3 - ( - 1 )} = \frac{- 8}{4} = - 2$

The gradient is $m = - 2$ . The negative sign tells us the line slopes downward from left to right.

Warning

Common mistake: Students sometimes subtract the coordinates in different orders — for example, $y_{2} - y_{1}$ on top but $x_{1} - x_{2}$ on the bottom. Always be consistent! If you start with point 2 on top, use point 2 first on the bottom too.

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