GCSE revision notes

This section runs through the key points associated with the EQUATION OF A LINE

Make sure you understand EVERY ASPECT covered. It is a source of particularly easy marks in the examination. DO NOT SKIP through any section.


 

Key skill 1:

 

The basic equation of a line – how to plot a line

 

Suppose you are asked to draw the line with the equation

 

Start by sketching the following table in which we choose three x-coordinates. The simplest ones are

 

 

 

 

 

Now work out what the  is for EACH of the  by using the given equation

 

When ,       

 

When ,       

 

When ,       

 

We get the following table:

 

 

Now plot the coordinates, remembering the  is the horizontal axis and the  is the vertical axis AND DRAW A LINE THROUGH THE POINTS

 

 

 

 

 


 

Key skill 2:

 

Negatives in the equation

 

Suppose you are asked to draw the line with the equation

 

Again, start by sketching the following table in which we choose three x-coordinates. The simplest ones are

 

 

 

 

 

Now work out what the  is for EACH of the  by using the given equation

 

When ,       

 

When ,       

 

When ,       

 

We get the following table:

 

 

Now plot the coordinates, remembering the  is the horizontal axis and the  is the vertical axis AND DRAW A LINE THROUGH THE POINTS

 

 

 

 

 

 

 

 

 

 

 

 

Straight lines and gradients

Any equation of the form                           is always a straight-line.

A concise summary of gradients

 

So,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 1:

So, for example  represents a straight line with a gradient of 2 and a y-intercept of 3.

The gradient of 2 means that for every 1 square across you go up 2 squares.

The y-intercept means it passes through 3 on the y-axis

 

Below are various examples of different gradients, including some more difficult ones

Example 1:

 

The gradient above equals 3 since it goes across 1 and up 3.

The y-intercept is -1 because it crosses through -1 on the y-axis

The equation is therefore

 

Example 2:

The gradient above is -2 because it goes across 1 and down 2

The y-intercept is +1 because it goes through +1 on the y-axis

The equation is therefore


Example 3:

A more difficult gradient

This line goes across 4 up 3.

The y-intercept is +1


Parallel lines

Parallel lines always have the same gradient. These are examples of pairs of parallel lines

 

 

It doesn’t matter what you add or subtract as long as the number before the x is the same

 

Perpendicular Gradients

To find a perpendicular gradient: he sign changes and you find the “RECIPROCAL”

So, to find the equation of a line perpendicular to

This lines has a gradient of

So the perpendicular line has a gradient of

 

Re-arranging equations – a grade A* idea