Equation of a Line

What you'll learn

How to read the gradient and y-intercept of a straight line.
How to use the form $y = m x + c$ .
How to find an equation from a graph or from key information.
How to recognise parallel and horizontal lines.

Coordinates and the y-axis

A coordinate grid has an $x$ -axis going across and a $y$ -axis going up. Points are written as pairs, like (3, 4).

A coordinate pair gives the horizontal position first and the vertical position second.

Definition

Coordinate pair

A coordinate pair tells you the $x$ -coordinate first, then the $y$ -coordinate. So (3, 4) means 3 across and 4 up.

A line crosses the $y$ -axis when its $x$ -coordinate is 0. This is a very common one-mark question.

Example

Crossing the y-axis

The line is $y = 4 x - 6$ . A line crosses the $y$ -axis where $x = 0$ .

The line crosses the y-axis at the point where x is 0, which is the y-intercept.

Substitute $x = 0$ into the equation:
$y = 4 (0) - 6 = - 6$
The value of $y$ where the line crosses the $y$ -axis is -6.

Gradient: how steep the line is

Definition

Gradient

The gradient of a straight line tells you its steepness. It is the change in $y$ divided by the change in $x$ , written as $\frac{\text{change in } y}{\text{change in } x}$ .

A positive gradient goes up as you move from left to right. A negative gradient goes down.

Positive gradients rise from left to right, while negative gradients fall from left to right.

Example

Finding a gradient from two points

Choose two clear points on the line, such as (1, 7) and (3, 1).

The gradient is found by comparing the vertical change with the horizontal change between two points on the line.

From the first point to the second, $x$ increases by 2 and $y$ decreases by 6.
Work out the gradient:
$\frac{-6}{2} = -3$
The gradient is -3, so the line slopes downwards from left to right.

Tip

Direction check

Always read gradients from left to right. Up means positive gradient; down means negative gradient.

The form $y = m x + c$

The most useful straight-line form at GCSE is:

y = m x + c

Definition

y-intercept

The y-intercept is where a line crosses the $y$ -axis. In $y = m x + c$ , the value of $c$ is the $y$ -intercept, so the crossing point is $(0, c)$ .

Key Idea

What $m$ and $c$ mean

In $y = m x + c$ , $m$ is the gradient and $c$ is the value where the line crosses the $y$ -axis.

In y = mx + c, the gradient controls the steepness and c gives the y-intercept.

Example

Reading $y = 6 - 2 x$

Rewrite the equation in the usual order:
$y = - 2 x + 6$
The number multiplying $x$ is -2, so the gradient is -2.
The number on its own is 6, so the line crosses the $y$ -axis at (0, 6).

Common Mistake

Forgetting the hidden 1

In $y = 2 - x$ , the $x$ term is really $- 1 x$ , so the gradient is -1, not 2 or 1.

Writing an equation from gradient and intercept

If you know the gradient and where the line crosses the $y$ -axis, you can put them straight into $y = m x + c$ .

Example

Using a gradient and a y-intercept

The line has gradient 4, so $m = 4$ .
It passes through (0, -3), so it crosses the $y$ -axis at -3. This means $c = - 3$ .
Substitute into $y = m x + c$ :
$y = 4 x - 3$

Finding the equation from a graph

To find a line’s equation from a graph, you need two things:

the gradient
the y-intercept

If the graph clearly crosses the $y$ -axis, read that point first.

Example

Finding an equation from two graph points

The line goes through (0, 5) and (2, -1). Since (0, 5) is on the $y$ -axis, $c = 5$ .

The point on the y-axis gives c, and the step between the two points gives the gradient.

From (0, 5) to (2, -1), $x$ increases by 2 and $y$ decreases by 6.
Work out the gradient:
$\frac{-6}{2} = -3$
Put $m = - 3$ and $c = 5$ into $y = m x + c$ :
$y = - 3 x + 5$

Rearranging into $y = m x + c$

Sometimes the equation is not already in the form $y = m x + c$ . The subject of an equation is the variable on its own. So “make $y$ the subject” means rearrange until it starts with $y =$ .

Example

Finding the gradient after rearranging

Start with $3 y - 6 x = 12$ . Add $6 x$ to both sides:
$3y−6x=123y=6x+12\begin{aligned} 3y - 6x &= 12 \\ 3y &= 6x + 12 \end{aligned}$
Divide every term by 3:
$y = 2 x + 4$
Now compare with $y = m x + c$ . The gradient is 2.

Parallel lines

Definition

Parallel lines

Parallel lines are lines that stay the same distance apart and never meet. Straight parallel lines have the same gradient.

Parallel straight lines have equal steepness, so they share the same gradient.

So if two lines are parallel, keep the same $m$ value. The $c$ value can change.

Example

Finding a parallel line

Rearrange $x + 2 y = 10$ into $y = m x + c$ :
$x+2y=102y=10−xy=−12x+5\begin{aligned} x + 2y &= 10 \\ 2y &= 10 - x \\ y &= -\frac{1}{2}x + 5 \end{aligned}$
The gradient is $−12-\frac{1}{2}$ , so a parallel line must also have gradient $−12-\frac{1}{2}$ .
The new line passes through (0, 1), so its y-intercept is 1.
The equation is $-\frac{1}{2}x + 1$ .

Matching equations to graphs

When matching lines to equations, look for quick clues:

Does it go up or down?
Where does it cross the $y$ -axis?
Is it horizontal?

A horizontal line is flat. Its equation looks like $y = c$ , because the $y$ -value stays the same.

A horizontal line has the same y-value everywhere, so its equation is y = c.

Example

Matching lines to equations

A positive line through the origin matches $y = 2 x$ , because there is no added constant.
A flat line crossing the $y$ -axis at 4 matches $y = 4$ .
A positive line crossing the $y$ -axis at 4 matches $y = 2 x + 4$ .
A negative line crossing the $y$ -axis at 4 matches $y = 4 - 2 x$ .

Exam technique

In the exam

Put the equation into $y = m x + c$ before reading the gradient.
On a graph, choose two exact grid points and use change in $y$ over change in $x$ .
For parallel lines, keep the gradient the same and change the intercept if needed.

Self review

Check yourself

Can you read the gradient and y-intercept from $y = 7 - 4 x$ ?
If a line goes through (0, -6) and has gradient 3, what would its equation look like?
Why do parallel lines have the same $m$ value in $y = m x + c$ ?

Equation of a Line

What you'll learn

How to read the gradient and y-intercept of a straight line.
How to use the form $y = m x + c$ .
How to find an equation from a graph or from key information.
How to recognise parallel and horizontal lines.

Coordinates and the y-axis

A coordinate grid has an $x$ -axis going across and a $y$ -axis going up. Points are written as pairs, like (3, 4).

A coordinate pair gives the horizontal position first and the vertical position second.

Definition

Coordinate pair

A coordinate pair tells you the $x$ -coordinate first, then the $y$ -coordinate. So (3, 4) means 3 across and 4 up.

A line crosses the $y$ -axis when its $x$ -coordinate is 0. This is a very common one-mark question.

Example

Crossing the y-axis

The line is $y = 4 x - 6$ . A line crosses the $y$ -axis where $x = 0$ .

The line crosses the y-axis at the point where x is 0, which is the y-intercept.

Substitute $x = 0$ into the equation:
$y = 4 (0) - 6 = - 6$
The value of $y$ where the line crosses the $y$ -axis is -6.

Gradient: how steep the line is

Definition

Gradient

The gradient of a straight line tells you its steepness. It is the change in $y$ divided by the change in $x$ , written as $\frac{\text{change in } y}{\text{change in } x}$ .

A positive gradient goes up as you move from left to right. A negative gradient goes down.

Positive gradients rise from left to right, while negative gradients fall from left to right.

Example

Finding a gradient from two points

Choose two clear points on the line, such as (1, 7) and (3, 1).

The gradient is found by comparing the vertical change with the horizontal change between two points on the line.

From the first point to the second, $x$ increases by 2 and $y$ decreases by 6.
Work out the gradient:
$\frac{-6}{2} = -3$
The gradient is -3, so the line slopes downwards from left to right.

Tip

Direction check

Always read gradients from left to right. Up means positive gradient; down means negative gradient.

The form $y = m x + c$

The most useful straight-line form at GCSE is:

y = m x + c

Definition

y-intercept

The y-intercept is where a line crosses the $y$ -axis. In $y = m x + c$ , the value of $c$ is the $y$ -intercept, so the crossing point is $(0, c)$ .

Key Idea

What $m$ and $c$ mean

In $y = m x + c$ , $m$ is the gradient and $c$ is the value where the line crosses the $y$ -axis.

In y = mx + c, the gradient controls the steepness and c gives the y-intercept.

Example

Reading $y = 6 - 2 x$

Rewrite the equation in the usual order:
$y = - 2 x + 6$
The number multiplying $x$ is -2, so the gradient is -2.
The number on its own is 6, so the line crosses the $y$ -axis at (0, 6).

Common Mistake

Forgetting the hidden 1

In $y = 2 - x$ , the $x$ term is really $- 1 x$ , so the gradient is -1, not 2 or 1.

Writing an equation from gradient and intercept

If you know the gradient and where the line crosses the $y$ -axis, you can put them straight into $y = m x + c$ .

Example

Using a gradient and a y-intercept

The line has gradient 4, so $m = 4$ .
It passes through (0, -3), so it crosses the $y$ -axis at -3. This means $c = - 3$ .
Substitute into $y = m x + c$ :
$y = 4 x - 3$

Finding the equation from a graph

To find a line’s equation from a graph, you need two things:

the gradient
the y-intercept

If the graph clearly crosses the $y$ -axis, read that point first.

Example

Finding an equation from two graph points

The line goes through (0, 5) and (2, -1). Since (0, 5) is on the $y$ -axis, $c = 5$ .

The point on the y-axis gives c, and the step between the two points gives the gradient.

From (0, 5) to (2, -1), $x$ increases by 2 and $y$ decreases by 6.
Work out the gradient:
$\frac{-6}{2} = -3$
Put $m = - 3$ and $c = 5$ into $y = m x + c$ :
$y = - 3 x + 5$

Rearranging into $y = m x + c$

Sometimes the equation is not already in the form $y = m x + c$ . The subject of an equation is the variable on its own. So “make $y$ the subject” means rearrange until it starts with $y =$ .

Example

Finding the gradient after rearranging

Start with $3 y - 6 x = 12$ . Add $6 x$ to both sides:
$3y−6x=123y=6x+12\begin{aligned} 3y - 6x &= 12 \\ 3y &= 6x + 12 \end{aligned}$
Divide every term by 3:
$y = 2 x + 4$
Now compare with $y = m x + c$ . The gradient is 2.

Parallel lines

Definition

Parallel lines

Parallel lines are lines that stay the same distance apart and never meet. Straight parallel lines have the same gradient.

Parallel straight lines have equal steepness, so they share the same gradient.

So if two lines are parallel, keep the same $m$ value. The $c$ value can change.

Example

Finding a parallel line

Rearrange $x + 2 y = 10$ into $y = m x + c$ :
$x+2y=102y=10−xy=−12x+5\begin{aligned} x + 2y &= 10 \\ 2y &= 10 - x \\ y &= -\frac{1}{2}x + 5 \end{aligned}$
The gradient is $−12-\frac{1}{2}$ , so a parallel line must also have gradient $−12-\frac{1}{2}$ .
The new line passes through (0, 1), so its y-intercept is 1.
The equation is $-\frac{1}{2}x + 1$ .

Matching equations to graphs

When matching lines to equations, look for quick clues:

Does it go up or down?
Where does it cross the $y$ -axis?
Is it horizontal?

A horizontal line is flat. Its equation looks like $y = c$ , because the $y$ -value stays the same.

A horizontal line has the same y-value everywhere, so its equation is y = c.

Example

Matching lines to equations

A positive line through the origin matches $y = 2 x$ , because there is no added constant.
A flat line crossing the $y$ -axis at 4 matches $y = 4$ .
A positive line crossing the $y$ -axis at 4 matches $y = 2 x + 4$ .
A negative line crossing the $y$ -axis at 4 matches $y = 4 - 2 x$ .

Exam technique

In the exam

Put the equation into $y = m x + c$ before reading the gradient.
On a graph, choose two exact grid points and use change in $y$ over change in $x$ .
For parallel lines, keep the gradient the same and change the intercept if needed.

Self review

Check yourself

Can you read the gradient and y-intercept from $y = 7 - 4 x$ ?
If a line goes through (0, -6) and has gradient 3, what would its equation look like?
Why do parallel lines have the same $m$ value in $y = m x + c$ ?

Equation of a Line

What you'll learn

Coordinates and the y-axis

Gradient: how steep the line is

The form $y = m x + c$

Writing an equation from gradient and intercept

Finding the equation from a graph

Rearranging into $y = m x + c$

Parallel lines

Matching equations to graphs

Recap questions

You've reached the end

Equation of a Line

What you'll learn

Coordinates and the y-axis

Gradient: how steep the line is

The form $y = m x + c$

Writing an equation from gradient and intercept

Finding the equation from a graph

Rearranging into $y = m x + c$

Parallel lines

Matching equations to graphs

Recap questions

You've reached the end

Equation of a Line

What you'll learn

Coordinates and the y-axis

Gradient: how steep the line is

The form y=mx+cy = mx + cy=mx+c

Writing an equation from gradient and intercept

Finding the equation from a graph

Rearranging into y=mx+cy = mx + cy=mx+c

Parallel lines

Matching equations to graphs

Recap questions

You've reached the end

Equation of a Line Revision Guide

Equation of a Line

What you'll learn

Coordinates and the y-axis

Gradient: how steep the line is

The form y=mx+cy = mx + cy=mx+c

Writing an equation from gradient and intercept

Finding the equation from a graph

Rearranging into y=mx+cy = mx + cy=mx+c

Parallel lines

Matching equations to graphs

Recap questions

You've reached the end

Equation of a Line Revision Guide

The form $y = m x + c$

Rearranging into $y = m x + c$

The form $y = m x + c$

Rearranging into $y = m x + c$