Gradient

Video lesson3 worked examples

Theory

The gradient (m) represents the steepness of a slope and is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Lines that are parallel have equal gradients.

You must be able to recognise and interpret lines with a zero gradient (a perfectly horizontal line) and gradients that are undefined (a perfectly vertical line).

The Golden Rule: subtract the coordinates in the same order on the top and bottom of the fraction. Equal gradients mean the lines are parallel — so to prove two lines parallel, show their gradients are equal and say so.

⚠️ Common Examiner Traps

Order consistency: if you write $y_2 - y_1$ on top you must write $x_2 - x_1$ on the bottom, in the same order — swapping one flips the sign.
Zero vs undefined: a horizontal line has gradient $0$ ; a vertical line's gradient is undefined (division by zero), not 0.
Negatives: subtracting a negative coordinate adds, e.g. $11 - (-5) = 16$ .
“Prove parallel” needs a statement: after showing the gradients are equal, write the conclusion — equal gradients alone earn nothing without it.

Worked examples

Example 1

Calculating Gradient

Find the gradient of the straight line passing through the points A(2, −5) and B(6, 11).

Step 1: Assign (x₁, y₁) to A and (x₂, y₂) to B.

Step 2: Substitute into the formula: $m = \frac{11 - (-5)}{6 - 2}$ .

Step 3: Simplify: $m = \frac{16}{4} = 4$ .

Answer: The gradient is 4.

Example 2

Parallel Lines

Line L₁ passes through (0, 4) and (3, 10). Line L₂ passes through (−1, 2) and (1, k). If L₁ and L₂ are parallel, find k.

Step 1: Find the gradient of L₁: $m = \frac{10 - 4}{3 - 0} = 2$ .

Step 2: Because they are parallel, L₂ also has a gradient of 2. Set up the equation for L₂: $\frac{k - 2}{1 - (-1)} = 2$ .

Step 3: Solve for k: $\frac{k - 2}{2} = 2 \rightarrow k - 2 = 4 \rightarrow k = 6$ .

Answer: k = 6.

Example 3

🔗 Bringing it together

The points are A $(-1, 2)$ , B $(2, 8)$ , C $(0, -3)$ and D $(3, 3)$ . Prove that AB is parallel to CD.

Step 1: Find the gradient of AB: $m_{AB} = \dfrac{8 - 2}{2 - (-1)} = \dfrac{6}{3} = 2$ .

Step 2: Find the gradient of CD, subtracting in the same order: $m_{CD} = \dfrac{3 - (-3)}{3 - 0} = \dfrac{6}{3} = 2$ .

Step 3: State the conclusion clearly — this is the part that earns the mark:

Answer: Since $m_{AB} = m_{CD} = 2$ , the lines AB and CD are parallel.

Ready to practise?

8 Gradient questions with answers and video solutions