Gradient of a line
The gradient (slope) measures how steep a line is: the vertical change divided by the horizontal change between two points. For points (x₁, y₁) and (x₂, y₂):
Key idea
gradient m = (y₂ − y₁) ÷ (x₂ − x₁). A positive m rises to the right; a negative m falls to the right.
The equation y = mx + c
Every straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept (the y-value where the line crosses the y-axis, i.e. where x = 0). Parallel lines share the same gradient m.
Worked example
A line passes through (2, 3) and (6, 11). Find its gradient and its equation.
Gradient m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
Use y = mx + c with the point (2, 3): 3 = 2 × 2 + c, so 3 = 4 + c, giving c = −1.
Equation: y = 2x − 1.
Intercepts
The y-intercept is c. To find the x-intercept, set y = 0 and solve for x. For y = 2x − 6, put y = 0: 0 = 2x − 6, so x = 3 — the line crosses the x-axis at (3, 0).
Remember
- m = (change in y) ÷ (change in x).
- In y = mx + c, m is the gradient and c is the y-intercept.
- Parallel lines → equal gradients. Set y = 0 to find where a line crosses the x-axis.