Equation of a Line

What you'll learn

How to read the gradient and y-intercept from a straight-line equation.
How to write equations in the form $y = m x + c$ .
How to find the equation of a line from a graph or from two points.
How to recognise and write equations of parallel lines.

Coordinates and straight lines: the basics

A coordinate grid has two axes:

the x-axis, which goes left and right
the y-axis, which goes up and down

A point is written as $(x, y)$ . The first number tells you how far across to go, and the second number tells you how far up or down to go.

Definition

Origin

The origin is the point where the x-axis and y-axis meet. Its coordinates are $(0, 0)$ .

Straight-line graphs often have equations involving $x$ and $y$ . For example, $y = 2 x + 3$ means: choose an $x$ value, multiply it by 2, then add 3 to get the matching $y$ value.

The form $y = m x + c$

The most useful form for a straight-line equation is:

y = m x + c

Definition

Gradient and y-intercept

In $y = m x + c$ :

$m$ is the gradient, which tells you the steepness and direction of the line.
$c$ is the y-intercept, which is where the line crosses the y-axis.

A coordinate graph showing y = 2x + 3, its y-intercept, a gradient triangle, and a parallel line

A positive gradient slopes upwards as you move from left to right. A negative gradient slopes downwards.

Key Idea

The main idea

If a line is written as $y = m x + c$ , you can read off the gradient and y-intercept immediately.

Example

Reading the gradient and y-intercept

A straight line has equation $y = 4 x - 6$ . Write down its gradient and the point where it crosses the y-axis.

Compare the equation with $y = m x + c$ .
$y = 4 x - 6$
The number multiplying $x$ is the gradient, so the gradient is 4.
The constant term is the y-intercept, so the line crosses the y-axis when $y = - 6$ .
The point on the y-axis has $x = 0$ , so the coordinate is $(0, - 6)$ .

Common Mistake

Forgetting the coordinate of the y-intercept

The y-intercept is not just the number -6. As a point, it is $(0, - 6)$ because every point on the y-axis has $x = 0$ .

Writing an equation from a gradient and y-intercept

If you know the gradient and the point where the line crosses the y-axis, you can build the equation straight away.

Use:

y = m x + c

where $m$ is the gradient and $c$ is the y-intercept.

Example

Writing the equation from a y-intercept

A line has gradient 3 and passes through $(0, - 4)$ . Find its equation.

The gradient is 3, so $m = 3$ .
The line passes through $(0, - 4)$ , so the y-intercept is -4. This means $c = - 4$ .
Substitute into $y = m x + c$ .
$y = 3 x - 4$

Tip

Quick check

If the point given has coordinates $(0, c)$ , it is already the y-intercept. You do not need to substitute it into a long formula.

Equations written in a slightly different order

Sometimes the equation is not written as $y = m x + c$ straight away.

For example:

y = 7 - 2 x

This is the same as:

y = - 2 x + 7

So the gradient is -2 and the y-intercept is 7.

Example

Reading an equation with the terms swapped

A straight line has equation $y = 5 - x$ . Find the gradient and the coordinates of the y-intercept.

Rewrite the equation so the $x$ term comes first.
$y = - x + 5$
The coefficient of $x$ is -1, so the gradient is -1.
The constant term is 5, so the line crosses the y-axis at $(0, 5)$ .

Common Mistake

Missing the hidden 1

In $y = 5 - x$ , the term $- x$ means $- 1 x$ . So the gradient is -1, not just “negative”.

Rearranging into $y = m x + c$

Some equations are given in forms like:

2 y - 6 x = 10

To find the gradient, first make $y$ the subject. That means rearrange the equation until $y$ is on its own.

Definition

Make y the subject

To make $y$ the subject means to rearrange an equation so it begins $y=…y=\dots$ .

Example

Rearranging to find the gradient

A straight line has equation $3 y - 12 x = 9$ . Find its gradient.

Start with the equation.
$3 y - 12 x = 9$
Add $12 x$ to both sides to move the $x$ term away from the left-hand side.
$3 y = 12 x + 9$
Divide every term by 3.
$y = 4 x + 3$
Now compare with $y = m x + c$ . The gradient is 4.

Example

Rearranging when x is added

A line has equation $x + 4 y = 12$ . Rearrange it to make $y$ the subject, then write down the gradient.

Start with the equation.
$x + 4 y = 12$
Subtract $x$ from both sides.
$4 y = 12 - x$
Divide every term by 4.
$y=3−14xy=3-\frac{1}{4}x$
Write it in $y = m x + c$ order.
$y=−14x+3y=-\frac{1}{4}x+3$
The gradient is $−14-\frac{1}{4}$ .

Common Mistake

Divide every term

When dividing by the number in front of $y$ , divide every term on the right-hand side as well. For example, from $4 y = 12 - x$ , you get $y=3−14xy=3-\frac{1}{4}x$ .

Finding the gradient from two points

If you can see two points on a line, you can calculate the gradient.

Definition

Gradient formula

The gradient between two points is:

x\text{gradient}=\frac{\text{change in }y}{\text{change in }x}

This is often remembered as “rise over run”.

For points $x_1,y_1)$ and $x_2,y_2)$ , the formula is:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Example

Finding an equation from two points

A straight line passes through $A (0, 2)$ and $B (3, 8)$ . Find the equation of the line.

Find the change in $y$ from $A$ to $B$ .
$8 - 2 = 6$
Find the change in $x$ from $A$ to $B$ .
$3 - 0 = 3$
Calculate the gradient.
$m=63=2m=\frac{6}{3}=2$
Since the line passes through $(0, 2)$ , the y-intercept is 2.
Put $m = 2$ and $c = 2$ into $y = m x + c$ .
$y = 2 x + 2$

Tip

Use the point where x is zero

If one of your points is on the y-axis, such as $(0, 2)$ , it gives you $c$ immediately. That saves time.

Finding the equation from a graph

When a straight line is drawn on a grid, use two clear grid points.

Choose points where the line goes exactly through a grid crossing. Avoid guessing halfway points unless the graph clearly shows them.

Example

Finding an equation from a drawn line

A line passes through the points $(0, 5)$ and $(2, - 1)$ . Find its equation.

The point $(0, 5)$ is on the y-axis, so the y-intercept is 5.
Find the change in $y$ from $(0, 5)$ to $(2, - 1)$ .
$- 1 - 5 = - 6$
Find the change in $x$ .
$2 - 0 = 2$
Calculate the gradient.
$m=−62=−3m=\frac{-6}{2}=-3$
Use $y = m x + c$ with $m = - 3$ and $c = 5$ .
$y = - 3 x + 5$

Common Mistake

Getting the sign of the gradient wrong

If the line goes down as you move from left to right, the gradient must be negative. This is a useful sanity check.

Parallel lines

Definition

Parallel lines

Parallel lines are straight lines that never meet. They have the same gradient.

Parallel lines can have different y-intercepts. For example, $y = 2 x + 1$ and $y = 2 x - 5$ are parallel because both have gradient 2.

Example

Writing the equation of a parallel line

A line is parallel to $y = - 3 x + 4$ and crosses the y-axis at $(0, 7)$ . Find its equation.

Parallel lines have the same gradient.
The gradient of $y = - 3 x + 4$ is -3.
The new line crosses the y-axis at $(0, 7)$ , so $c = 7$ .
Substitute into $y = m x + c$ .
$y = - 3 x + 7$

Example

Parallel line after rearranging first

A line has equation $2 y - 8 x = 6$ . Write down the equation of another line parallel to it.

Rearrange the equation to make $y$ the subject.
$2 y = 8 x + 6$
Divide every term by 2.
$y = 4 x + 3$
The gradient is 4, so any parallel line must also have gradient 4.
Choose a different y-intercept, for example -2.
$y = 4 x - 2$

Tip

Many answers can be correct

If a question asks for “another line parallel to” a given line, there are many possible answers. Keep the same gradient but choose a different y-intercept.

Matching equations to lines

To match equations with lines on a graph, look for two quick clues:

the direction of the slope: positive, negative, or horizontal
the y-intercept: where the line crosses the y-axis

A horizontal line has equation $y = c$ , where $c$ is a constant. For example, $y = 3$ is a horizontal line through every point with y-coordinate 3.

Example

Matching a line to its equation

A graph shows four lines. One line slopes downwards and crosses the y-axis at 4. Which equation matches it: $y = 2 x$ , $y = 4 - 3 x$ , $y = 3 x + 4$ , or $y = 4$ ?

The line slopes downwards, so its gradient is negative.
The line crosses the y-axis at 4, so $c = 4$ .
Look for the equation with negative gradient and y-intercept 4.
The matching equation is:
$y = 4 - 3 x$

Exam technique

In the exam

First check whether the equation is already in $y = m x + c$ form. If not, rearrange it before reading the gradient.
For a graph question, choose two clear grid points and calculate “change in $y$ over change in $x$ ”.
For parallel lines, copy the gradient exactly and change only the y-intercept if you need a different line.

Self review

Check yourself

If a line has equation $y = - 5 x + 2$ , what are its gradient and y-intercept?
How would you rearrange $2 x + 3 y = 12$ to make $y$ the subject?
What must be the same for two straight lines to be parallel?

Equation of a Line

What you'll learn

How to read the gradient and y-intercept from a straight-line equation.
How to write equations in the form $y = m x + c$ .
How to find the equation of a line from a graph or from two points.
How to recognise and write equations of parallel lines.

Coordinates and straight lines: the basics

A coordinate grid has two axes:

the x-axis, which goes left and right
the y-axis, which goes up and down

A point is written as $(x, y)$ . The first number tells you how far across to go, and the second number tells you how far up or down to go.

Definition

Origin

The origin is the point where the x-axis and y-axis meet. Its coordinates are $(0, 0)$ .

Straight-line graphs often have equations involving $x$ and $y$ . For example, $y = 2 x + 3$ means: choose an $x$ value, multiply it by 2, then add 3 to get the matching $y$ value.

The form $y = m x + c$

The most useful form for a straight-line equation is:

y = m x + c

Definition

Gradient and y-intercept

In $y = m x + c$ :

$m$ is the gradient, which tells you the steepness and direction of the line.
$c$ is the y-intercept, which is where the line crosses the y-axis.

A coordinate graph showing y = 2x + 3, its y-intercept, a gradient triangle, and a parallel line

A positive gradient slopes upwards as you move from left to right. A negative gradient slopes downwards.

Key Idea

The main idea

If a line is written as $y = m x + c$ , you can read off the gradient and y-intercept immediately.

Example

Reading the gradient and y-intercept

A straight line has equation $y = 4 x - 6$ . Write down its gradient and the point where it crosses the y-axis.

Compare the equation with $y = m x + c$ .
$y = 4 x - 6$
The number multiplying $x$ is the gradient, so the gradient is 4.
The constant term is the y-intercept, so the line crosses the y-axis when $y = - 6$ .
The point on the y-axis has $x = 0$ , so the coordinate is $(0, - 6)$ .

Common Mistake

Forgetting the coordinate of the y-intercept

The y-intercept is not just the number -6. As a point, it is $(0, - 6)$ because every point on the y-axis has $x = 0$ .

Writing an equation from a gradient and y-intercept

If you know the gradient and the point where the line crosses the y-axis, you can build the equation straight away.

Use:

y = m x + c

where $m$ is the gradient and $c$ is the y-intercept.

Example

Writing the equation from a y-intercept

A line has gradient 3 and passes through $(0, - 4)$ . Find its equation.

The gradient is 3, so $m = 3$ .
The line passes through $(0, - 4)$ , so the y-intercept is -4. This means $c = - 4$ .
Substitute into $y = m x + c$ .
$y = 3 x - 4$

Tip

Quick check

If the point given has coordinates $(0, c)$ , it is already the y-intercept. You do not need to substitute it into a long formula.

Equations written in a slightly different order

Sometimes the equation is not written as $y = m x + c$ straight away.

For example:

y = 7 - 2 x

This is the same as:

y = - 2 x + 7

So the gradient is -2 and the y-intercept is 7.

Example

Reading an equation with the terms swapped

A straight line has equation $y = 5 - x$ . Find the gradient and the coordinates of the y-intercept.

Rewrite the equation so the $x$ term comes first.
$y = - x + 5$
The coefficient of $x$ is -1, so the gradient is -1.
The constant term is 5, so the line crosses the y-axis at $(0, 5)$ .

Common Mistake

Missing the hidden 1

In $y = 5 - x$ , the term $- x$ means $- 1 x$ . So the gradient is -1, not just “negative”.

Rearranging into $y = m x + c$

Some equations are given in forms like:

2 y - 6 x = 10

To find the gradient, first make $y$ the subject. That means rearrange the equation until $y$ is on its own.

Definition

Make y the subject

To make $y$ the subject means to rearrange an equation so it begins $y=…y=\dots$ .

Example

Rearranging to find the gradient

A straight line has equation $3 y - 12 x = 9$ . Find its gradient.

Start with the equation.
$3 y - 12 x = 9$
Add $12 x$ to both sides to move the $x$ term away from the left-hand side.
$3 y = 12 x + 9$
Divide every term by 3.
$y = 4 x + 3$
Now compare with $y = m x + c$ . The gradient is 4.

Example

Rearranging when x is added

A line has equation $x + 4 y = 12$ . Rearrange it to make $y$ the subject, then write down the gradient.

Start with the equation.
$x + 4 y = 12$
Subtract $x$ from both sides.
$4 y = 12 - x$
Divide every term by 4.
$y=3−14xy=3-\frac{1}{4}x$
Write it in $y = m x + c$ order.
$y=−14x+3y=-\frac{1}{4}x+3$
The gradient is $−14-\frac{1}{4}$ .

Common Mistake

Divide every term

When dividing by the number in front of $y$ , divide every term on the right-hand side as well. For example, from $4 y = 12 - x$ , you get $y=3−14xy=3-\frac{1}{4}x$ .

Finding the gradient from two points

If you can see two points on a line, you can calculate the gradient.

Definition

Gradient formula

The gradient between two points is:

x\text{gradient}=\frac{\text{change in }y}{\text{change in }x}

This is often remembered as “rise over run”.

For points $x_1,y_1)$ and $x_2,y_2)$ , the formula is:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Example

Finding an equation from two points

A straight line passes through $A (0, 2)$ and $B (3, 8)$ . Find the equation of the line.

Find the change in $y$ from $A$ to $B$ .
$8 - 2 = 6$
Find the change in $x$ from $A$ to $B$ .
$3 - 0 = 3$
Calculate the gradient.
$m=63=2m=\frac{6}{3}=2$
Since the line passes through $(0, 2)$ , the y-intercept is 2.
Put $m = 2$ and $c = 2$ into $y = m x + c$ .
$y = 2 x + 2$

Tip

Use the point where x is zero

If one of your points is on the y-axis, such as $(0, 2)$ , it gives you $c$ immediately. That saves time.

Finding the equation from a graph

When a straight line is drawn on a grid, use two clear grid points.

Choose points where the line goes exactly through a grid crossing. Avoid guessing halfway points unless the graph clearly shows them.

Example

Finding an equation from a drawn line

A line passes through the points $(0, 5)$ and $(2, - 1)$ . Find its equation.

The point $(0, 5)$ is on the y-axis, so the y-intercept is 5.
Find the change in $y$ from $(0, 5)$ to $(2, - 1)$ .
$- 1 - 5 = - 6$
Find the change in $x$ .
$2 - 0 = 2$
Calculate the gradient.
$m=−62=−3m=\frac{-6}{2}=-3$
Use $y = m x + c$ with $m = - 3$ and $c = 5$ .
$y = - 3 x + 5$

Common Mistake

Getting the sign of the gradient wrong

If the line goes down as you move from left to right, the gradient must be negative. This is a useful sanity check.

Parallel lines

Definition

Parallel lines

Parallel lines are straight lines that never meet. They have the same gradient.

Parallel lines can have different y-intercepts. For example, $y = 2 x + 1$ and $y = 2 x - 5$ are parallel because both have gradient 2.

Example

Writing the equation of a parallel line

A line is parallel to $y = - 3 x + 4$ and crosses the y-axis at $(0, 7)$ . Find its equation.

Parallel lines have the same gradient.
The gradient of $y = - 3 x + 4$ is -3.
The new line crosses the y-axis at $(0, 7)$ , so $c = 7$ .
Substitute into $y = m x + c$ .
$y = - 3 x + 7$

Example

Parallel line after rearranging first

A line has equation $2 y - 8 x = 6$ . Write down the equation of another line parallel to it.

Rearrange the equation to make $y$ the subject.
$2 y = 8 x + 6$
Divide every term by 2.
$y = 4 x + 3$
The gradient is 4, so any parallel line must also have gradient 4.
Choose a different y-intercept, for example -2.
$y = 4 x - 2$

Tip

Many answers can be correct

If a question asks for “another line parallel to” a given line, there are many possible answers. Keep the same gradient but choose a different y-intercept.

Matching equations to lines

To match equations with lines on a graph, look for two quick clues:

the direction of the slope: positive, negative, or horizontal
the y-intercept: where the line crosses the y-axis

A horizontal line has equation $y = c$ , where $c$ is a constant. For example, $y = 3$ is a horizontal line through every point with y-coordinate 3.

Example

Matching a line to its equation

A graph shows four lines. One line slopes downwards and crosses the y-axis at 4. Which equation matches it: $y = 2 x$ , $y = 4 - 3 x$ , $y = 3 x + 4$ , or $y = 4$ ?

The line slopes downwards, so its gradient is negative.
The line crosses the y-axis at 4, so $c = 4$ .
Look for the equation with negative gradient and y-intercept 4.
The matching equation is:
$y = 4 - 3 x$

Exam technique

In the exam

First check whether the equation is already in $y = m x + c$ form. If not, rearrange it before reading the gradient.
For a graph question, choose two clear grid points and calculate “change in $y$ over change in $x$ ”.
For parallel lines, copy the gradient exactly and change only the y-intercept if you need a different line.

Self review

Check yourself

If a line has equation $y = - 5 x + 2$ , what are its gradient and y-intercept?
How would you rearrange $2 x + 3 y = 12$ to make $y$ the subject?
What must be the same for two straight lines to be parallel?

Equation of a Line

What you'll learn

Coordinates and straight lines: the basics

The form $y = m x + c$

Writing an equation from a gradient and y-intercept

Equations written in a slightly different order

Rearranging into $y = m x + c$

Finding the gradient from two points

Finding the equation from a graph

Parallel lines

Matching equations to lines

Recap questions

You've reached the end

Equation of a Line

What you'll learn

Coordinates and straight lines: the basics

The form $y = m x + c$

Writing an equation from a gradient and y-intercept

Equations written in a slightly different order

Rearranging into $y = m x + c$

Finding the gradient from two points

Finding the equation from a graph

Parallel lines

Matching equations to lines

Recap questions

You've reached the end

Equation of a Line

What you'll learn

Coordinates and straight lines: the basics

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

Writing an equation from a gradient and y-intercept

Equations written in a slightly different order

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

Finding the gradient from two points

Finding the equation from a graph

Parallel lines

Matching equations to lines

Recap questions

You've reached the end

Equation of a Line Revision Guide

Equation of a Line

What you'll learn

Coordinates and straight lines: the basics

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

Writing an equation from a gradient and y-intercept

Equations written in a slightly different order

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

Finding the gradient from two points

Finding the equation from a graph

Parallel lines

Matching equations to lines

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 $y = m x + c$