GCSE MATHS • HIGHER TIER

Equation of a Line Made Simple

Understand y = mx + c, identify gradients and intercepts, draw straight-line graphs and find equations from graphs or coordinates.

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Equation of a Line in Four Moves

1

Identify

Write the line in the form y = mx + c whenever possible.

2

Read

Identify m as the gradient and c as the y-intercept.

3

Calculate

Use coordinates, a table of values or the gradient formula when needed.

4

Check

Confirm that the equation matches the graph, points and intercept.

What You’ll Learn

Understand the form y = mx + c.

Identify the gradient and y-intercept from an equation.

Create a table of values for a straight-line graph.

Draw straight-line graphs accurately.

Find the equation of a line from its gradient and intercept.

Find the equation of a line from a graph or two coordinates.

Recognise horizontal and vertical lines.

Key Rules

Line equation formula

y = mx + c where m = gradient and c = y-intercept.

Gradient

Gradient shows how steep the line is. Calculate it using Gradient = Change in y ÷ Change in x.

Y-intercept

The y-intercept is the point where the line crosses the y-axis.

Finding an equation

Identify the gradient (m) and intercept (c), then substitute them into y = mx + c.

Worked Proof: Equation of a Line

Step-by-Step Method

  • 1. Write the equation in the form y = mx + c.
  • 2. Find the gradient.
  • 3. Find the y-intercept.
  • 4. Substitute the values into y = mx + c.
  • 5. Check the equation.

More Worked Examples

1

State the gradient and intercept of:

y = 5x + 4

Step 1: Compare with y = mx + c.
Step 2: Identify the coefficient of x.
Gradient = 5
Step 3: Identify the constant term.
Intercept = 4
Answer: Gradient = 5, Intercept = 4

2

State the gradient and intercept of:

y = -2x + 7

Step 1: The coefficient of x is -2.
Gradient = -2
Step 2: The constant term is 7.
Intercept = 7
Answer: Gradient = -2, Intercept = 7

3

Creating a Table of Values

y = x + 3

x = 0 → y = 0 + 3 = 3
x = 1 → y = 1 + 3 = 4
x = 2 → y = 2 + 3 = 5
Coordinates: (0, 3), (1, 4), (2, 5)

4

Draw the graph of:

y = 2x + 1

Step 1: Create a table.
(0, 1), (1, 3), (2, 5), (3, 7)
Step 2: Plot the coordinates.
Step 3:Join the points with a straight line.
  • Common GCSE Mistakes
  • Mixing up gradient and intercept.
  • Forgetting negative signs.
  • Using incorrect coordinates.
  • Plotting points inaccurately.
  • Drawing curved lines instead of straight lines.
  • Exam Tips
  • Write y = mx + c before identifying the gradient and intercept.
  • Remember that m is the coefficient of x and c is the y-intercept.
  • Keep negative gradients and negative intercepts clearly signed.
  • When drawing a graph, plot coordinates carefully and join them with a straight line.
  • For a line through two points, find m first and then substitute a point to find c.

Skills Used in Equation of a Line

Linear Graphs

Gradient

Y-Intercept

Coordinates

Tables of Values

Straight-Line Equations

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Frequently Asked Questions

It is the standard form of a straight-line equation, where m is the gradient and c is the y-intercept.

The gradient measures the steepness of a line and can be calculated as change in y divided by change in x.

It is the y-value where the line crosses the y-axis.

Choose x-values, calculate the corresponding y-values, plot the coordinates and join them with a straight line.

Calculate the gradient first, then substitute one point into y = mx + c to find c.

Horizontal lines have equations such as y = 4 and gradient 0. Vertical lines have equations such as x = 5 and have no gradient.