A Level Maths: The Equation Of A Straight Line Part 1
Example
The equation of a straight line is taught at both GCSE and A Level. The concepts that were introduced at GCSE are carried forward such as being able to determine the gradient, midpoint, length and also the equation of a straight line.
The equation of a straight line is in the form y = mx + c where m is the gradient and c is where the straight line crosses the y-axis.
Finding the Gradient
Given two points on a line you can determine the gradient by dividing the vertical height by the horizontal distance. You may have heard of this as “rise over run”. Remember the gradient measures the steepness of a line and it can have a positive or negative gradient.
The mathematical expression for the gradient of a straight line is given by:
What you need to be careful about is that you are finding “the change in y over the change in x” and not the other way round.
Also you need to be consistent in what you are using in terms of coordinates. From the diagram it does not matter if you start with coordinate A or coordinate B. However if you start with A then these are your 
coordinates. Do not then pick B to also be a part of 
.
Example
Find the equation of the straight line joining the points P(5, 3) and Q(-7, 12)
When it comes to geometry you need to get into the habit of drawing a diagram regardless of how simple you think the question is. It does not need to be accurate but a simple sketch can help you to visualise the question more easily.
To find the gradient, determine the starting point. Remember can start with either but be consistent.
For the above calculation Q was the starting point.
Let us see what happens if P is the starting point.
The same answer is obtained. For the second version, notice the double negative in the denominator.
Example
What you need to be ready to do is to be ready to solve questions that have a slight “twist” to them such as the following:
P(2, -5) and Q(4, a) form part of a straight line with gradient -1. Find the value of a.
Here you are not being asked to find a gradient, you are asked to find out a missing coordinate. Here the key is to look for the keywords within the question and in this case it is the word gradient.
We know:
Inserting the values for what we know:
You will see similar problems to this but it is a case of using what you already know. We have still used the rule to determine the gradient in order to answer the question.
There will be other terms that you will also meet such as “collinear” which just means that points all lie on the same straight line. A common question is to be given three coordinates and to determine a missing value just as per the last example.
With two of the points you can find the gradient of the line and once that is known you can then use the coordinate with the missing value and any other point to determine the missing value. It then becomes the same question as above.
The equation of a straight line is a very common question and whether you are doing AQA, OCR, MEI or Edexcel A Level Maths Revision, you need to focus on more awkwards questions.
We will continue with more information regarding straight line geometry which is applicable to A Level Maths for year 12 students.
Different Forms For The Equation Of A Straight Line
The general equation of a straight line is written in the form y = mx + c and this can also be written in the form ax + by + c = 0.
Make sure that you are reading the question carefully in terms of how a question wants you to present an answer.
Another important skill that is needed is to be able to write down the gradient of a straight line. It most cases this is simply the coefficient of x. But there will be equations where you need to make y the subject.
For instance: what is the gradient of the following straight line?
5x + 3y – 7 = 0
This is not in the general form of y = mx + c and it needs to be. Making y the subject gives:
So the gradient is: 
Sketching A Straight Line
At GCSE level in order to draw a straight line you were generally asked to complete a table of values, plot the points and then connect all the points with a straight line.
Whilst during A Level Maths you will rarely find an instance where you have to draw a straight line or a curve to scale. A sketch means just that – a sketch – where all that is required is that you are clearly labelling any points of intersection with the axes.
Example
Sketch the straight line y = 7x + 2
In order to sketch a straight line there are two important facts to consider:
Along the x-axis the value of y is always 0.
Which means the line will cross the x-axis at:
Along the y-axis the value of x is always 0.
Which means that line will cross the y-axis at:
y = 2
You now have all the information that you need to sketch the straight line:
As you can from the diagram, by no means, is this drawn to scale. All we are interested in is the general shape and where it is crossing the axes.
Exam Style Question:
Regardless how simple you think the question is, always draw a sketch.
From our sketch we have a download slope which means that the gradient is negative and this agrees with our calculation.
The most commonly used general equation of a straight line is y = mx + c
You can use this to find the equation of the straight line. You know the gradient and you know that it passes through two given points. Pick one of the points and substitute this in. The aim is to find the value of “c”.
The equation of the straight line is:
BUT, this is not what the question wants. Although the answer is correct, the format of the answer is not correct. Always be looking at questions as they do ask for answers in a certain way.
To obtain the format that the question wants, simply multiply throughout by 5 to give:
There will be additional articles for the equation of a straight line in due course as well as looking at circle geometry.
Overall this is generally a not-so-complicated area of A Level Maths but what is important is that you are drawing a sketch. Geometry is such a visual topic that without a diagram the question can be very difficult to answer.
Our 3 and 4 day A Level Maths Revision Course in London examines in detail some of the more challenging questions and how best to answer them in order to achieve full marks. All courses are popular but Easter and May are very popular due to the run up to the final summer exams.
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