1.5.6 Draw a line given its equation

What you'll learn

How to recognise the equation of a straight line.
How to use the gradient and intercept to plot a line efficiently.
How to rearrange equations that are not already in a convenient form.
How to draw horizontal and vertical lines, including lines with fractional gradients.

Coordinates and straight lines

A point on a coordinate grid is written as an ordered pair $(x, y)$ . The first coordinate gives the horizontal position and the second gives the vertical position.

For example, the point $(3, - 2)$ is 3 units to the right of the origin and 2 units below it.

The origin is the point $(0, 0)$ , where the coordinate axes meet. The horizontal axis is the $x$ -axis, and the vertical axis is the $y$ -axis.

Definition

Equation of a line

An equation of a line describes the relationship between the $x$ - and $y$ -coordinates of every point on that line. A point lies on the line precisely when its coordinates satisfy the equation.

For example, $(2, 5)$ lies on the line $y = 2 x + 1$ because substituting $x = 2$ gives

y = 2 (2) + 1 = 5.

However, $(2, 4)$ does not lie on the line because its coordinates do not satisfy the equation.

The form $y = m x + c$

The most useful form of a straight-line equation is

y = m x + c .

Definition

Gradient

The gradient, represented by $m$ , measures the steepness and direction of a line:

x.m=\frac{\text{change in }y}{\text{change in }x}.

It is also described as the rise divided by the run.

A positive gradient means the line rises from left to right. A negative gradient means it falls from left to right. A gradient of zero gives a horizontal line.

Definition

y-intercept

The $y$ -intercept, represented by $c$ , is the $y$ -coordinate where the line crosses the $y$ -axis. At this point, $x = 0$ , so its coordinates are $(0, c)$ .

For the line $y = 2 x - 1$ :

the gradient is $m = 2$ ;
the $y$ -intercept is $c = - 1$ ;
the line crosses the $y$ -axis at $(0, - 1)$ .

Key Idea

Reading a line equation

When an equation is written as $y = m x + c$ , the coefficient of $x$ is the gradient and the constant term is the $y$ -intercept.

Example

Drawing a line from its gradient and intercept

Draw the line $y = 2 x - 1$ .

Identify the $y$ -intercept: $c = - 1$ , so plot the point $(0, - 1)$ .
Interpret the gradient as $2=212=\frac{2}{1}$ . From $(0, - 1)$ , move 1 unit to the right and 2 units upwards. This gives the point $(1, 1)$ .
Repeat the same movement to obtain another point, $(2, 3)$ . Alternatively, move 1 unit left and 2 units down to obtain $(- 1, - 3)$ .
Use a ruler to draw one straight line through the points, extending it across the relevant coordinate grid.

Common Mistake

Reversing the gradient

For $m=23m=\frac{2}{3}$ , the vertical change is 2 and the horizontal change is 3. Do not move 3 units vertically and 2 units horizontally.

Using a table of values

You can always draw a line by choosing some $x$ -values, calculating the corresponding $y$ -values, and plotting the resulting coordinates.

This method is especially useful when the gradient or intercept is not immediately obvious.

Example

Generating points from an equation

Draw the line $y = - 3 x + 4$ by calculating three points.

Choose convenient values such as $x = - 1$ , $x = 0$ and $x = 1$ .
Substitute each value into the equation:

x=−1:y=−3(−1)+4=7,x=0:y=−3(0)+4=4,x=1:y=−3(1)+4=1.\begin{aligned} x=-1 &: \quad y=-3(-1)+4=7,\\ x=0 &: \quad y=-3(0)+4=4,\\ x=1 &: \quad y=-3(1)+4=1. \end{aligned}

Plot the points $(- 1, 7)$ , $(0, 4)$ and $(1, 1)$ .
Draw a straight line through the points. Its downward direction agrees with the negative gradient $m = - 3$ .

Tip

Choose useful values

Choose values that keep the coordinates within the given grid. For a fractional gradient, carefully chosen multiples of the denominator often produce integer coordinates.

Fractional gradients

A fractional gradient tells you directly how to move between points.

For example,

m=32m=\frac{3}{2}

means a rise of 3 units for a run of 2 units. A negative gradient such as

m=−32m=-\frac{3}{2}

can be treated as a fall of 3 units while moving 2 units to the right.

Example

Using a fractional gradient

Draw the line $y=−23x+3y=-\frac{2}{3}x+3$ .

Plot the $y$ -intercept $(0, 3)$ .
Read the gradient as $−23-\frac{2}{3}$ . From $(0, 3)$ , move 3 units to the right and 2 units down, reaching $(3, 1)$ .
Repeat the movement to reach $(6, - 1)$ , or reverse it by moving 3 units left and 2 units up to reach $(- 3, 5)$ .
Draw a straight line through the plotted points.

Rearranging an equation first

A linear equation may not initially be written as $y = m x + c$ . You can rearrange it by making $y$ the subject, meaning that $y$ is isolated on one side.

Example

Rearranging before drawing

Draw the line $2 x + 3 y = 12$ .

Rearrange the equation to make $y$ the subject:

2x+3y=123y=12−2xy=4−23xy=−23x+4.\begin{aligned} 2x+3y&=12\\ 3y&=12-2x\\ y&=4-\frac{2}{3}x\\ y&=-\frac{2}{3}x+4. \end{aligned}

The $y$ -intercept is 4, so plot $(0, 4)$ .
The gradient is $−23-\frac{2}{3}$ . Move 3 units right and 2 units down to obtain $(3, 2)$ , then repeat to obtain $(6, 0)$ .
Draw the line through these points. The point $(6, 0)$ also shows that the line crosses the $x$ -axis at 6.

Common Mistake

Dividing only one term

When dividing $3 y = 12 - 2 x$ by 3, divide every term on the right-hand side. The result is $y=4−23xy=4-\frac{2}{3}x$ , not $y = 4 - 2 x$ .

Using the intercepts

An $x$ -intercept is a point where a graph crosses the $x$ -axis. Every point on the $x$ -axis has $y = 0$ .

To find the two axis intercepts:

set $x = 0$ to find the $y$ -intercept;
set $y = 0$ to find the $x$ -intercept.

Two distinct points determine a unique straight line, so the two intercepts are often enough to draw it.

Example

Drawing a line from its intercepts

Draw the line $4 x + 3 y = 12$ .

Set $x = 0$ to find the $y$ -intercept:

\Rightarrow y=4,

so one point is $(0, 4)$ .

Set $y = 0$ to find the $x$ -intercept:

\Rightarrow x=3,

so the other point is $(3, 0)$ .

Plot $(0, 4)$ and $(3, 0)$ , then draw the straight line through them.

Tip

Choose the quickest method

For an equation such as $4 x + 3 y = 12$ , finding the intercepts may be quicker than rearranging. For $y = 5 x - 2$ , reading the gradient and $y$ -intercept is usually quicker.

Horizontal and vertical lines

A horizontal line has a constant $y$ -coordinate. Its equation has the form

y = k,

where $k$ is a constant. For example, $y = 3$ is a horizontal line through every point whose $y$ -coordinate is 3.

Its gradient is zero because there is no change in $y$ .

A vertical line has a constant $x$ -coordinate. Its equation has the form

x = k .

For example, $x = - 2$ is a vertical line through every point whose $x$ -coordinate is $- 2$ .

Common Mistake

Vertical lines

A vertical line cannot be written as $y = m x + c$ . Its gradient is undefined because the horizontal change is zero, which would require division by zero.

Example

Drawing horizontal and vertical lines

Draw $y = - 1$ and $x = 4$ on the same coordinate grid.

For $y = - 1$ , plot points such as $(- 2, - 1)$ and $(3, - 1)$ . Join them with a horizontal line.
For $x = 4$ , plot points such as $(4, - 2)$ and $(4, 3)$ . Join them with a vertical line.
The lines meet at $(4, - 1)$ because this point satisfies both equations.

Checking your graph

Substitute the coordinates of a plotted point into the original equation. If the two sides are equal, the point is on the line.

You should also check the overall direction:

positive gradient: rises from left to right;
negative gradient: falls from left to right;
zero gradient: horizontal;
equation $x = k$ : vertical.

Exam technique

In the exam

Rearrange into $y = m x + c$ if this makes the gradient and intercept easy to identify.
Plot at least two accurate points; use a third point as a check when practical.
Draw one straight, continuous line with a ruler and extend it across the required grid.
Check the sign of the gradient and substitute one plotted point into the original equation.

Self review

Check yourself

How would you draw the line $y=12x−3y=\frac{1}{2}x-3$ using its gradient and intercept?
What are the two axis intercepts of $2 x + 5 y = 10$ ?
Why can the line $x = 6$ not be written in the form $y = m x + c$ ?

What you'll learn

How to recognise the equation of a straight line.
How to use the gradient and intercept to plot a line efficiently.
How to rearrange equations that are not already in a convenient form.
How to draw horizontal and vertical lines, including lines with fractional gradients.

Coordinates and straight lines

A point on a coordinate grid is written as an ordered pair $(x, y)$ . The first coordinate gives the horizontal position and the second gives the vertical position.

For example, the point $(3, - 2)$ is 3 units to the right of the origin and 2 units below it.

The origin is the point $(0, 0)$ , where the coordinate axes meet. The horizontal axis is the $x$ -axis, and the vertical axis is the $y$ -axis.

Definition

Equation of a line

An equation of a line describes the relationship between the $x$ - and $y$ -coordinates of every point on that line. A point lies on the line precisely when its coordinates satisfy the equation.

For example, $(2, 5)$ lies on the line $y = 2 x + 1$ because substituting $x = 2$ gives

y = 2 (2) + 1 = 5.

However, $(2, 4)$ does not lie on the line because its coordinates do not satisfy the equation.

The form $y = m x + c$

The most useful form of a straight-line equation is

y = m x + c .

Definition

Gradient

The gradient, represented by $m$ , measures the steepness and direction of a line:

x.m=\frac{\text{change in }y}{\text{change in }x}.

It is also described as the rise divided by the run.

A positive gradient means the line rises from left to right. A negative gradient means it falls from left to right. A gradient of zero gives a horizontal line.

Definition

y-intercept

The $y$ -intercept, represented by $c$ , is the $y$ -coordinate where the line crosses the $y$ -axis. At this point, $x = 0$ , so its coordinates are $(0, c)$ .

For the line $y = 2 x - 1$ :

the gradient is $m = 2$ ;
the $y$ -intercept is $c = - 1$ ;
the line crosses the $y$ -axis at $(0, - 1)$ .

Key Idea

Reading a line equation

When an equation is written as $y = m x + c$ , the coefficient of $x$ is the gradient and the constant term is the $y$ -intercept.

Example

Drawing a line from its gradient and intercept

Draw the line $y = 2 x - 1$ .

Identify the $y$ -intercept: $c = - 1$ , so plot the point $(0, - 1)$ .
Interpret the gradient as $2=212=\frac{2}{1}$ . From $(0, - 1)$ , move 1 unit to the right and 2 units upwards. This gives the point $(1, 1)$ .
Repeat the same movement to obtain another point, $(2, 3)$ . Alternatively, move 1 unit left and 2 units down to obtain $(- 1, - 3)$ .
Use a ruler to draw one straight line through the points, extending it across the relevant coordinate grid.

Common Mistake

Reversing the gradient

For $m=23m=\frac{2}{3}$ , the vertical change is 2 and the horizontal change is 3. Do not move 3 units vertically and 2 units horizontally.

Using a table of values

You can always draw a line by choosing some $x$ -values, calculating the corresponding $y$ -values, and plotting the resulting coordinates.

This method is especially useful when the gradient or intercept is not immediately obvious.

Example

Generating points from an equation

Draw the line $y = - 3 x + 4$ by calculating three points.

Choose convenient values such as $x = - 1$ , $x = 0$ and $x = 1$ .
Substitute each value into the equation:

x=−1:y=−3(−1)+4=7,x=0:y=−3(0)+4=4,x=1:y=−3(1)+4=1.\begin{aligned} x=-1 &: \quad y=-3(-1)+4=7,\\ x=0 &: \quad y=-3(0)+4=4,\\ x=1 &: \quad y=-3(1)+4=1. \end{aligned}

Plot the points $(- 1, 7)$ , $(0, 4)$ and $(1, 1)$ .
Draw a straight line through the points. Its downward direction agrees with the negative gradient $m = - 3$ .

Tip

Choose useful values

Choose values that keep the coordinates within the given grid. For a fractional gradient, carefully chosen multiples of the denominator often produce integer coordinates.

Fractional gradients

A fractional gradient tells you directly how to move between points.

For example,

m=32m=\frac{3}{2}

means a rise of 3 units for a run of 2 units. A negative gradient such as

m=−32m=-\frac{3}{2}

can be treated as a fall of 3 units while moving 2 units to the right.

Example

Using a fractional gradient

Draw the line $y=−23x+3y=-\frac{2}{3}x+3$ .

Plot the $y$ -intercept $(0, 3)$ .
Read the gradient as $−23-\frac{2}{3}$ . From $(0, 3)$ , move 3 units to the right and 2 units down, reaching $(3, 1)$ .
Repeat the movement to reach $(6, - 1)$ , or reverse it by moving 3 units left and 2 units up to reach $(- 3, 5)$ .
Draw a straight line through the plotted points.

Rearranging an equation first

A linear equation may not initially be written as $y = m x + c$ . You can rearrange it by making $y$ the subject, meaning that $y$ is isolated on one side.

Example

Rearranging before drawing

Draw the line $2 x + 3 y = 12$ .

Rearrange the equation to make $y$ the subject:

2x+3y=123y=12−2xy=4−23xy=−23x+4.\begin{aligned} 2x+3y&=12\\ 3y&=12-2x\\ y&=4-\frac{2}{3}x\\ y&=-\frac{2}{3}x+4. \end{aligned}

The $y$ -intercept is 4, so plot $(0, 4)$ .
The gradient is $−23-\frac{2}{3}$ . Move 3 units right and 2 units down to obtain $(3, 2)$ , then repeat to obtain $(6, 0)$ .
Draw the line through these points. The point $(6, 0)$ also shows that the line crosses the $x$ -axis at 6.

Common Mistake

Dividing only one term

When dividing $3 y = 12 - 2 x$ by 3, divide every term on the right-hand side. The result is $y=4−23xy=4-\frac{2}{3}x$ , not $y = 4 - 2 x$ .

Using the intercepts

An $x$ -intercept is a point where a graph crosses the $x$ -axis. Every point on the $x$ -axis has $y = 0$ .

To find the two axis intercepts:

set $x = 0$ to find the $y$ -intercept;
set $y = 0$ to find the $x$ -intercept.

Two distinct points determine a unique straight line, so the two intercepts are often enough to draw it.

Example

Drawing a line from its intercepts

Draw the line $4 x + 3 y = 12$ .

Set $x = 0$ to find the $y$ -intercept:

\Rightarrow y=4,

so one point is $(0, 4)$ .

Set $y = 0$ to find the $x$ -intercept:

\Rightarrow x=3,

so the other point is $(3, 0)$ .

Plot $(0, 4)$ and $(3, 0)$ , then draw the straight line through them.

Tip

Choose the quickest method

For an equation such as $4 x + 3 y = 12$ , finding the intercepts may be quicker than rearranging. For $y = 5 x - 2$ , reading the gradient and $y$ -intercept is usually quicker.

Horizontal and vertical lines

A horizontal line has a constant $y$ -coordinate. Its equation has the form

y = k,

where $k$ is a constant. For example, $y = 3$ is a horizontal line through every point whose $y$ -coordinate is 3.

Its gradient is zero because there is no change in $y$ .

A vertical line has a constant $x$ -coordinate. Its equation has the form

x = k .

For example, $x = - 2$ is a vertical line through every point whose $x$ -coordinate is $- 2$ .

Common Mistake

Vertical lines

A vertical line cannot be written as $y = m x + c$ . Its gradient is undefined because the horizontal change is zero, which would require division by zero.

Example

Drawing horizontal and vertical lines

Draw $y = - 1$ and $x = 4$ on the same coordinate grid.

For $y = - 1$ , plot points such as $(- 2, - 1)$ and $(3, - 1)$ . Join them with a horizontal line.
For $x = 4$ , plot points such as $(4, - 2)$ and $(4, 3)$ . Join them with a vertical line.
The lines meet at $(4, - 1)$ because this point satisfies both equations.

Checking your graph

Substitute the coordinates of a plotted point into the original equation. If the two sides are equal, the point is on the line.

You should also check the overall direction:

positive gradient: rises from left to right;
negative gradient: falls from left to right;
zero gradient: horizontal;
equation $x = k$ : vertical.

Exam technique

In the exam

Rearrange into $y = m x + c$ if this makes the gradient and intercept easy to identify.
Plot at least two accurate points; use a third point as a check when practical.
Draw one straight, continuous line with a ruler and extend it across the required grid.
Check the sign of the gradient and substitute one plotted point into the original equation.

Self review

Check yourself

How would you draw the line $y=12x−3y=\frac{1}{2}x-3$ using its gradient and intercept?
What are the two axis intercepts of $2 x + 5 y = 10$ ?
Why can the line $x = 6$ not be written in the form $y = m x + c$ ?

What you'll learn

Coordinates and straight lines

The form $y = m x + c$

Using a table of values

Fractional gradients

Rearranging an equation first

Using the intercepts

Horizontal and vertical lines

Checking your graph

Review 1.5.6 Draw a line given its equation by teaching Genie

1.5.6 Draw a line given its equation

What you'll learn

Coordinates and straight lines

The form $y = m x + c$

Using a table of values

Fractional gradients

Rearranging an equation first

Using the intercepts

Horizontal and vertical lines

Checking your graph

Review 1.5.6 Draw a line given its equation by teaching Genie

1.5.6 Draw a line given its equation

What you'll learn

Coordinates and straight lines

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

Using a table of values

Fractional gradients

Rearranging an equation first

Using the intercepts

Horizontal and vertical lines

Checking your graph

Review 1.5.6 Draw a line given its equation by teaching Genie

1.5.6 Draw a line given its equation Revision Guide

1.5.6 Draw a line given its equation

What you'll learn

Coordinates and straight lines

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

Using a table of values

Fractional gradients

Rearranging an equation first

Using the intercepts

Horizontal and vertical lines

Checking your graph

Review 1.5.6 Draw a line given its equation by teaching Genie

1.5.6 Draw a line given its equation Revision Guide

The form $y = m x + c$

The form $y = m x + c$