What is a linear equation?
A linear equation is a statement with an equal sign which declares that two expressions have equal values; for example, 
We use the word ‘linear’ because there is no power in the 
Equations that contain terms with 
Before tackling some examples, let’s summarise several algebraic laws into a single statement:
If we perform identical mathematical operations on both sides of the equal sign, the final result will not change.
What this means is that if the same number is added, subtracted, or multiplied on both sides of the equal sign the final result will be identical. As GCSE maths students should know, this is called balancing.
Broadly speaking, this is the method of attack: we want all the
Let’s get started with a few examples.
1a. We’ll start off with an easy one here.
Let’s separate and collect our
Therefore, 
1b. Let’s change it round and work with a subtraction
Again, we’ll separate out and collect our
Therefore, 
1c. Now we’re going to make the equations slightly more complex
Let’s separate and collect our
$4x-3x=4-2$
We can solve this through simple arithmetic to discover that
1d. Now it’s time to make things a bit harder by working with negative quantities
Let’s separate and collect our different terms:
Therefore 
You can always check your work by substituting the final result into the original equation. For instance, in example 1a the final result is 
However, sometimes things get more complicated and we need to multiply or divide both sides by the same number to find the solution.
2a. Let’s work with some multiplication and division
In this example, we want to multiply both sides by 2:
It’s now an easy step to diving 4 by 2 and finding that
2b. In this example, we’re going to multiply both sides by the denominator
We need to multiply them both out by 2.
Therefore we can work out that
2c. Let’s try a similar, but slightly different example
Here, we want to multiply both sides by 3.
We can now use simple arithmetic to find that
Sometimes, things can get a bit more complicated, and we’ll need to use other techniques such as expanding brackets.
In this final section, we are going to solve the kind of linear equations you’d usually find in GCSE Maths papers
3a. Here’s a classic GCSE Maths linear equation
We want to start by multiplying out the brackets.
Now let’s separate out and collect the numbers and the
With some fairly easy maths, we now have our answer:
3b. We’ll solve this one using the same method
Multiply out the brackets:
Separate and collect our numbers and our
This gives us:
Therefore 
3c. Now let’s look at a more complex linear equation that involves subtraction
Multiply out the brackets:
Solve and simplify:
Separate out and collect the numbers and
Simplify them down further:
We have our answer:
3d. Let’s do another one using the methods we’ll be very familiar with by now
Multiply out the brackets:
Separate out and collect the numbers and
Simplify:
We have a fractional answer:
3e. Try this one out
Multiply out the brackets:
Separate and collect our numbers and
Simplify:
Therefore:
3f. Let’s have one last go. You should be a natural by now!
Multiply out the brackets:
Simplify:
Separate out and collect:
Therefore
Really, the only way to learn how to solve algebraic linear equations is through a lot of practice. It’s a repetitive but straightforward process that shouldn’t take you too long to get used to.
I hope this article has helped you feel more confident in this fundamental area of GCSE maths algebra.
