Find the gradient/slope of a line

Finding the Gradient (Slope) of a Line

What is Gradient?

The gradient (also called slope) of a line tells us how steep the line is. It shows the ratio of the vertical change to the horizontal change between any two points on the line.

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 horizontal line has a gradient of zero, while a vertical line has an undefined gradient.

The Formula

For two points on a line with coordinates $A(x_1, y_1)$ and $B(x_2, y_2)$, the gradient $m$ is:

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

This formula compares the change in y (vertical change) to the change in x (horizontal change).

Steps to Find the Gradient

1. Identify two points on the line and label them as $(x_1, y_1)$ and $(x_2, y_2)$.
2. Subtract the first y-coordinate from the second: $y_2 - y_1$.
3. Subtract the first x-coordinate from the second: $x_2 - x_1$.
4. Divide the result of step 2 by the result of step 3.

Worked Example

Problem: A straight line passes through points $P(2, 3)$ and $Q(6, 11)$. Find the gradient of this line.

Solution:

Let $P(2, 3)$ be the first point so $x_1 = 2$, $y_1 = 3$.

Let $Q(6, 11)$ be the second point so $x_2 = 6$, $y_2 = 11$.

Now apply the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$

Therefore, the gradient of the line is 2. This means for every 1 unit the line moves to the right, it rises by 2 units.

Important Notes

• The order of points does not matter; you will get the same result if you swap them, as long as you keep the subtraction consistent.
• If the gradient is a fraction, leave it as a fraction (e.g., $\frac{3}{4}$) rather than converting to decimals.
• When $x_2 = x_1$, you are dividing by zero, which means the line is vertical and its gradient is undefined.

Real-life application

In Tanzania, gradient is used in construction and road building. For example, when engineers design roads in places like Dodoma or along the highway from Dar es Salaam to Morogoro, they must calculate the gradient of slopes to ensure vehicles can travel safely. A road that is too steep (high positive gradient) may be dangerous for trucks and buses, especially during the rainy season when roads can become slippery.