Linear graphs

The equation y = mx + c

Every straight line can be written as y = mx + c where:

• m = gradient (slope) — the change in y for every 1 unit increase in x.
• c = y-intercept — the value of y when x = 0 (where the line crosses the y-axis).

Finding the gradient from a graph: gradient = rise ÷ run = (change in y) ÷ (change in x). Pick two points on the line and calculate.

Finding the equation from two points: calculate the gradient, then substitute one point into y = mx + c to find c.

Example: Points (2, 5) and (6, 13). m = (13 − 5)/(6 − 2) = 8/4 = 2. 5 = 2(2) + c → c = 1. Equation: y = 2x + 1.

Parallel lines

Parallel lines have the same gradient. If a line has gradient m, any parallel line is y = mx + k for a different constant k.

Perpendicular lines

If a line has gradient m, a perpendicular line has gradient −1/m (the negative reciprocal).

Example: a line with gradient 3 is perpendicular to a line with gradient −1/3. Example: gradient 2/3 → perpendicular gradient = −3/2.

Check: m₁ × m₂ = −1 for perpendicular lines.

Finding the equation of a line through a given point

Use y − y₁ = m(x − x₁), where (x₁, y₁) is the known point.

Example: Line through (3, −1) with gradient 4: y − (−1) = 4(x − 3) → y + 1 = 4x − 12 → y = 4x − 13.

Edexcel exam style

Edexcel frequently tests:

• "Find the equation of the line perpendicular to L that passes through point P."
• "Show that two lines are parallel." (Show gradients are equal.)
• Interpret gradient and intercept in a real-world context (e.g. "what does the gradient represent?").