Sheet № 92 · Foundation + Higher · AQA · Edexcel · OCR
Finding the Equation of a Line –
Finding the equation of a straight line is one of the most tested algebra skills at GCSE. Whether you are given a gradient and a point, two points, or told the line is parallel or perpendicular to another, the goal is always the same: write the equation in the form y = mx + c.
§Key definitions
Question:
Find the equation of the line with gradient 3 that passes through (2, 11).
Answer:
y = 3x + 5
Q1 (Foundation):
Find the equation of the line with gradient 5 that passes through (1, 8).
Q2 (Foundation):
Find the equation of the line through (2, 3) and (6, 11).
Q3 (Higher):
A line is parallel to y = 4x + 1 and passes through (3, 5). Find its equation.
§Formulas to memorise
y = mx + c
Gradient = (y₂ − y₁) / (x₂ − x₁)
y = mx + c — substitute a known point to find c
Parallel lines have equal gradients: m₁ = m₂
Perpendicular lines have gradients that multiply to −1: m₁ × m₂ = −1
Write y = mx + c and substitute the given gradient for m.
Substitute the gradient and one of the points into y = mx + c.
Substitute the new gradient and the given point into y = mx + c, then solve for c.
Worked example
Find the equation of the line with gradient 3 that passes through (2, 11).
Working:
⚠ Common mistakes
- ✗Swapping the coordinates in the gradient formula. Always subtract in the same order: (y₂ − y₁) on top and (x₂ − x₁) on the bottom. Mixing up the order gives the wrong sign.
- ✗Forgetting the negative reciprocal for perpendicular lines. The perpendicular gradient is not just the reciprocal — you must also flip the sign. The gradient perpendicular to 3 is −1/3, not 1/3.
- ✗Leaving c out of the final answer. After finding c, make sure you write the full equation y = mx + c. Some students solve for c but never state the equation.
✦ Exam tips
- →Always show your substitution step — writing 11 = 3(2) + c earns method marks even if you make an arithmetic slip.
- →If the question says "parallel", the gradient stays the same. If it says "perpendicular", flip and negate.
- →Check your answer by substituting the given point back into your final equation to confirm it works.