Linear Graphs & Equation of a Line – GCSE Guide

Linear graphs are one of the most frequently tested topics across AQA, Edexcel and OCR GCSE Maths papers. Whether you are on Foundation or Higher tier, you need to plot straight lines, read off gradients, identify intercepts and write the equation of a line. This guide walks you through everything from the basics of y = mx + c to finding equations of parallel and perpendicular lines — a Higher-tier favourite worth up to five marks per question.

What Is a Linear Graph?

A linear graph is a straight-line graph. Every straight line can be described by an equation in the form:

y = mx + c

where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).

The gradient tells you how much y increases for every 1 unit increase in x. A positive gradient slopes upward from left to right; a negative gradient slopes downward.

Key formulas

Gradient between two points (x₁, y₁) and (x₂, y₂):

m = (y₂ − y₁) / (x₂ − x₁)

Parallel lines have the same gradient: m₁ = m₂

Perpendicular lines have gradients that multiply to −1:

m₁ × m₂ = −1

This means the perpendicular gradient is the negative reciprocal. If one line has gradient 3, the perpendicular gradient is −1/3.

Step-by-Step Method

How to plot a straight line from y = mx + c

Identify m and c from the equation. For example, y = 2x + 3 has m = 2 and c = 3.
Plot the y-intercept (0, c) on the y-axis.
Use the gradient to find a second point. Gradient 2 means go 1 right and 2 up.
Plot a third point the same way to confirm the line is straight.
Draw a straight line through the points using a ruler, extending it across the grid.

How to find the equation of a line from a graph

Read off the y-intercept — the value where the line crosses the y-axis. This gives c.
Pick two clear points on the line where it crosses grid intersections.
Calculate the gradient using m = (y₂ − y₁) / (x₂ − x₁).
Write the equation as y = mx + c.

How to find the equation given a point and a gradient

Substitute the gradient m and the coordinates of the point (x, y) into y = mx + c.
Solve for c.
Write the full equation y = mx + c.

Worked Example 1 — Foundation Level

Question: A line passes through (0, 4) and (3, 10). Find its equation.

Step 1: The line passes through (0, 4), so the y-intercept c = 4.

Step 2: Calculate the gradient.

m = (10 − 4) / (3 − 0) = 6 / 3 = 2

Step 3: Write the equation.

y = 2x + 4

Check: When x = 3, y = 2(3) + 4 = 10 ✓

Worked Example 2 — Higher Level

Question: Line L passes through (2, 7) and is perpendicular to the line y = −½x + 1. Find the equation of line L.

Step 1: The gradient of y = −½x + 1 is m = −½.

Step 2: The perpendicular gradient is the negative reciprocal:

m = −1 ÷ (−½) = 2

Step 3: Substitute m = 2 and the point (2, 7) into y = mx + c:

7 = 2(2) + c → 7 = 4 + c → c = 3

Step 4: The equation of line L is:

y = 2x + 3

Check: When x = 2, y = 2(2) + 3 = 7 ✓. The gradients multiply to (−½) × 2 = −1 ✓

Common Mistakes

Mixing up rise and run. The gradient is change in y divided by change in x, not the other way round. Always put the vertical change on top.
Forgetting the sign of the gradient. If the line slopes downward from left to right, the gradient is negative. Students often write it as positive.
Using the wrong reciprocal for perpendicular lines. The perpendicular gradient is the negative reciprocal, not just the reciprocal. If m = 2, the perpendicular gradient is −½, not ½.
Not reading the scale carefully. On exam graphs, the axes may not go up in ones. Always check what each square represents.
Writing x = instead of y =. The equation of a straight line is y = mx + c. Lines of the form x = k are vertical lines, which have undefined gradient.

Exam Tips

Show your gradient calculation. Write out (y₂ − y₁) / (x₂ − x₁) in full — examiners award method marks even if you make an arithmetic slip.
Label graphs clearly. When plotting, always label the line with its equation. On AQA papers in particular, the mark scheme often requires this.
Use the cover-up method for lines in the form ax + by = c. Set x = 0 to find where it crosses the y-axis, then set y = 0 for the x-axis crossing.
For perpendicular questions, write the word "negative reciprocal" in your working — it shows the examiner you understand the concept and can earn you an extra communication mark on Edexcel papers.

Practice Questions

Question 1 (Foundation): A straight line has the equation y = 3x − 5. State the gradient and the y-intercept.

Answer: Gradient = 3, y-intercept = −5 (the point (0, −5))

Question 2 (Foundation/Higher): Find the equation of the line passing through (1, 5) and (4, 14).

Answer: m = (14 − 5)/(4 − 1) = 9/3 = 3. Substituting (1, 5): 5 = 3(1) + c → c = 2. Equation: y = 3x + 2

Question 3 (Higher): Line A has equation y = 4x − 1. Line B is perpendicular to Line A and passes through (8, 3). Find the equation of Line B.

Answer: Perpendicular gradient = −¼. Substituting (8, 3): 3 = −¼(8) + c → 3 = −2 + c → c = 5. Equation: y = −¼x + 5

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Summary

A linear graph is a straight line described by y = mx + c, where m is the gradient and c is the y-intercept.
The gradient measures steepness: m = (y₂ − y₁) / (x₂ − x₁).
A positive gradient slopes upward; a negative gradient slopes downward.
Parallel lines share the same gradient.
Perpendicular lines have gradients that are negative reciprocals of each other (they multiply to −1).
To find the equation of a line, you need either two points or one point and the gradient.
Always check your equation by substituting a known point back in.

Linear Graphs & Equation of a Line –