Find the gradient of the line passing through the points P(1, 1) and Q(2, 5).

Analyze the question: We need the gradient (slope) of the straight line joining P(1, 1) and Q(2, 5).

Concept: The gradient of a line through two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ − y₁) / (x₂ − x₁)

This is "rise over run": how much y changes for each unit change in x.

Solution:

• (x₁, y₁) = (1, 1)
• (x₂, y₂) = (2, 5)

m = (5 − 1) / (2 − 1) m = 4 / 1 m = 4

Check: Moving from P to Q, x increases by 1 and y increases by 4. So the line rises 4 units for every 1 unit across, which confirms a gradient of 4.

Answer: The gradient is 4, which is option D.

Exam tip: Always subtract the coordinates in the same order (y₂ − y₁ over x₂ − x₁). If you flip one but not the other, you'll get the wrong sign. Also, a quick sanity check: if the denominator is 1, the gradient equals the difference in y-values directly.