# Equation Of A Line

Here we will learn about the equation of a line, including recognising the gradient and y-intercept of a straight line, and finding the equation of a line from a graph.

There are also worksheets on the equation of a line based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is the equation of a line?

The equation of a line is the algebraic representation of a line using cartesian coordinates.

The general form of the equation of a straight line is written as y = mx + c .
Where m is the gradient of the straight line
and c is the y -intercept of the straight line.

The coordinate ( x,y) lies on the line y = mx + c giving us a linear relationship between x and y . The term linear equation is given to any straight line.

E.g.
Let’s look at the line y = 3x − 4 .

Here, we can find any y -coordinate given the value for x by substituting into the equation y = 3x − 4 .

E.g.
When x = 1 ,

\begin{aligned} &y=3 \times 1-4 \\\\ &y=3-4 \\\\ &y=-1 \end{aligned}

So when the x value is 1 and the y value is -1 , this gives us the coordinate ( 1, -1 ) which lies on the line as shown below.

Not all straight lines will appear to be exactly in the form y = mx + c so we need to understand how we determine the gradient (or steepness of the line) m , and the y -intercept (the point where the line intersects the y -axis) c from equations that are not in the form y = mx + c .

Here are some examples of linear equations not in the form y = mx + c

• y + 17 = 6x
• 2y = 10x + 3
• x = 6y −1
• y = −3 (a horizontal line)
• x = y
• x = 0 (a vertical line)

In order to easily determine m and c we need to rearrange the equation to make y the subject.

E.g.
Take the equation above of y + 17 = 6x and make y the subject.

By rearranging the equation into the form y = mx + c we can clearly state that the gradient m = 6 and the y -intercept c = −17 .

From this, we can draw the straight line onto a set of axes.

## How to find the equation of a line

In order to find the equation of a straight line:

1. Calculate the gradient of the line
2. State the y -intercept of the straight line
3. Write the equation of the line in the form y = mx + c

### Related lessons on straight line graphs

Equation of a line is part of our series of lessons to support revision on straight line graphs. You may find it helpful to start with the main straight line graphs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Equation of a line examples

### Example 1: Positive gradient, positive y-intercept

Work out the equation of the straight line given in the diagram below.

1. Calculate the gradient of the line.

Two points that lie on the line are: ( 0, 4 ) and ( 2, 8 ).The line passes through these two given points.

So the gradient

\begin{aligned} &m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-4}{2-0}=\frac{4}{2}=2\\\\ &m=2 \end{aligned}

2State the y-intercept of the straight line.

The y -intercept occurs when x = 0 .

Here, c = 4

3Write the equation of the line in the form y = mx + c .

As the slope m = 2 and y intercept c = 4 ,

y=2x+4

### Example 2: Positive gradient, negative y-intercept

Work out the equation of the straight line given in the diagram below.

Calculate the gradient of the line.

State the y -intercept of the straight line.

Write the equation of the line in the form y=mx+c .

### Example 3: Negative gradient, positive y-intercept

Work out the equation of the straight line given in the diagram below.

Calculate the gradient of the line.

State the y -intercept of the straight line.

Write the equation of the line in the form y=mx+c .

### Example 4: Negative gradient, negative y-intercept

Work out the equation of the straight line given in the diagram below.

Calculate the gradient of the line.

State the y -intercept of the straight line.

Write the equation of the line in the form y=mx+c .

### Example 5: Positive fractional gradient, positive y-intercept

Work out the equation of the straight line given in the diagram below.

Calculate the gradient of the line.

State the y -intercept of the straight line.

Write the equation of the line in the form y=mx+c .

### Example 6: Negative fractional gradient, negative y-intercept

Work out the equation of the straight line given in the diagram below.

Calculate the gradient of the line.

State the y -intercept of the straight line.

Write the equation of the line in the form y=mx+c .

### Common misconceptions

• Mixing up the gradient and the y -intercept

The coefficient of x is the gradient m and the constant term is the y -intercept c . If the coefficient of x is 1 , remember this is written as x only as 1x = x .

• The gradient is positive or negative

Ignoring the negative values in a coordinate means that the gradient will be calculated incorrectly. When picking two coordinates, make sure that the coordinates go through the corner of a grid square. If there are negative values, make sure you use them with the negative symbol.

• Mixing up the coordinates

When calculating the gradient of a straight line, be careful to not mix up the coordinates.
E.g.
Looking back at example 5 , we have the two points that lie on the line being ( 1, −2 ) and ( −1,8 ).

The change in y could be correctly calculated as 8 \; − \; −2 but the change in x is then incorrectly calculated as 1 \; − \; −1 . This would result in the gradient of the line being 5 and not −5

Correct answer: m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8--2}{-1-1}=\frac{10}{-2}=-5

• Incorrect change in x or y

Avoid counting squares to work out the change in x or y . Use the axes scales or label the coordinates, then find the difference between them.

• Not calculating the gradient

A common error is to not calculate the gradient of the line.
Let’s look at example 1 ,

A error could be made by identifying the y -intercept and determining that the equation of the line is y = x + 4 .
This will only work when x = 0 but not for any other value for x .

### Practice equation of a line questions

1. Work out the equation of the straight line

y=x+5

y=5x+1

x+1

y=2.5x+1

Two coordinates: (0,1) and (2,11)

m=\frac{11-1}{2-0}=\frac{10}{2}=5

c=1

2. Work out the equation of the straight line

y=-8x+6

x-8

y=6x-8

y=3x-8

Two coordinates: (0,-8) and (1,-2)

m=\frac{-2-  -8}{1-0}=\frac{6}{1}=6

c=-8

3. Work out the equation of the straight line

y=-5x+10

y=10x-5

x+10

y=-2.5x+10

Two coordinates: (2,0) and (1,5)

m=\frac{5-0}{1-2}=\frac{5}{-1}=-5

c=10

4. Work out the equation of the straight line

y=x-1

x-1

y=-0.5x

y=-x

Two coordinates: (3,-3) and (4,-4)

m=\frac{-4-  -3}{4-3}=\frac{-1}{1}=-1

c=0

5. Work out the equation of the straight line

y=\frac{3}{2}x+2

x+2

y=\frac{2}{3}x+2

y=\frac{1}{3}x+2

Two coordinates: (0,2) and (3,4)

m=\frac{4-2}{3-0}=\frac{2}{3}

c=2

6 Work out the equation of the straight line

y=-\frac{1}{2}x-\frac{13}{2}

y=-\frac{13}{2}x-\frac{1}{2}

x-6.5

y=-\frac{1}{4}x-6.5

Two coordinates: (3,-8) and (-3,-5)

m=\frac{-5-  -8}{-3-3}=\frac{3}{-6}=-\frac{1}{2}

c=-\frac{13}{2}

### Equation of a line GCSE questions

1. Here is the graph of a straight line.

Work out the equation of the line in the form y=mx+c.

(3 marks)

Show answer

Two coordinates: (12,-2) and (0,6) and Gradient m=\frac{6-  -2}{0-12}=\frac{8}{-12}=-\frac{3}{4}

(1)

c=6

(1)

y=-\frac{3}{4}x+6

(1)

2.  (a)  Calculate the slope of the line in the diagram below.

(b) Show that the equation of the line is the same as \frac{1}{2}y=x+2.

(4 marks)

Show answer

(a)

Two coordinates: (0,4) and (2,8) and Gradient m=\frac{8-4}{2-0}=\frac{4}{2}=2

(1)

(b)

c=4

(1)

y=2x+4

(1)

(1)

3. The two straight lines A and B are parallel. The equation of line A is y=\frac{1}{2}x+7.

Given that equation B passes through the coordinate (4,3) , work out the equation of the straight line, B .

(4 marks)

Show answer

Parallel line so m=\frac{1}{2}

(1)

At (4,3), \; 3=4\times\frac{1}{2}+c

(1)

c=1

(1)

y=\frac{1}{2}x+1

(1)

## Learning checklist

You have now learned how to:

• Reduce a given linear equation in 2 variables to the standard form y = mx + c ; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

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