Gradient of a straight line
The gradient (slope) measures how steep a line is. Between two points (x₁, y₁) and (x₂, y₂) it equals the change in y divided by the change in x.
Key idea
gradient m = (y₂ - y₁) / (x₂ - x₁)
The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.
Parallel and perpendicular lines
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is -1, so if one gradient is m the other is -1/m. A vertical line (equal x-values) has an undefined gradient.
Worked example
Find the gradient of the line through (1, 2) and (3, 8), then its equation.
m = (8 - 2)/(3 - 1) = 6/2 = 3.
Using y = mx + c with point (1, 2): 2 = 3(1) + c, so c = -1.
The equation is y = 3x - 1.
Finding the equation
Once you know the gradient m and one point, substitute into y = mx + c to find c. To check whether a point lies on a line, substitute its coordinates: if both sides are equal, the point is on the line.
Remember
- Rearrange to the form y = mx + c to read off gradient and intercept.
- Parallel ⇒ same m; perpendicular ⇒ m₁ × m₂ = -1.
- A horizontal line has gradient 0; a vertical line has undefined gradient.