# How to Solve Linear Equations for GCSE Maths

## 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, $2x + 4$. Solving linear equations is a fundamental part of algebra at GCSE maths level, but it’s applicable to even some of the most advanced mathematical problems.

We use the word ‘linear’ because there is no power in the $x$ term. Well, $x$ itself is a power of one, but we don’t have to write that down.

Equations that contain terms with $x^{2}$ are quadratic ,$x^{3}$ are cubic, $x^{4}$ quartic etc.

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 $x$ terms on the left-hand side of the equal sign and all the numbers on the right-hand side. Because the equal sign acts like a “wall”, every time we move something from one side to the other we need to change its sign. The remaining part is just simple arithmetic.

Let’s get started with a few examples.

## 1a. We’ll start off with an easy one here.

$x+2=5$

Let’s separate and collect our $x$ terms and our numbers. Now we have

$x=5-2$

Therefore, $x=3$.

## 1b. Let’s change it round and work with a subtraction

$x-2=5$

Again, we’ll separate out and collect our $x$ terms and our numbers to give us:

$x=5+2$

Therefore, $x=7$.

## 1c. Now we’re going to make the equations slightly more complex

$4x+2=3x+4$

Let’s separate and collect our $x$ terms and our numbers to make:

\$4x-3x=4-2\$

We can solve this through simple arithmetic to discover that $x=2$

## 1d. Now it’s time to make things a bit harder by working with negative quantities

$-2x+3=-3x+4$

Let’s separate and collect our different terms:

$- 2x+3x=4-3$

Therefore $x=1$

You can always check your work by substituting the final result into the original equation. For instance, in example 1a the final result is $x=3$ . Substituting 3 into the original equation you get $3+2=5$ and $5=5$ which is an identity.

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

$2x=4$

In this example, we want to multiply both sides by 2:

$frac{2}{x} =frac{4}{2}f$

It’s now an easy step to diving 4 by 2 and finding that $x=1$.

## 2b. In this example, we’re going to multiply both sides by the denominator

$frac{1}{2} x = frac{3}{2}$

We need to multiply them both out by 2.

$( 2) frac{1}{2} x = frac{3}{2} ( 2)$

Therefore we can work out that $x=3$

## 2c. Let’s try a similar, but slightly different example

$frac{1}{3} x=6$

Here, we want to multiply both sides by 3.

$( 3)frac{1}{3} x=6( 3)$

We can now use simple arithmetic to find that $x=18$

Sometimes, things can get a bit more complicated, and we’ll need to use other techniques such as expanding brackets.

## 3a. Here’s a classic GCSE Maths linear equation

$4 ( x+1) =3 ( x+2)$

We want to start by multiplying out the brackets.

$4x+4=3x+6$

Now let’s separate out and collect the numbers and the $x$ terms.

$4x-3x=6-4$

With some fairly easy maths, we now have our answer:

$x=2$

## 3b. We’ll solve this one using the same method

$3( x+1) =x+4$

Multiply out the brackets:

$3x+3=x+4$

Separate and collect our numbers and our $x$ terms:

$3x-x=4-3$

This gives us:

$2x=1$

Therefore $x=frac{1}{2}$

## 3c. Now let’s look at a more complex linear equation that involves subtraction

$10-5x=3( x-4) -2( x+7)$

Multiply out the brackets:

$10-5x=3x-12-2x-14$

Solve and simplify:

$10-5x=x-26$

Separate out and collect the numbers and $x$ terms:

$-5x-x= -26-10$

Simplify them down further:

$-6x=-36$

$x=6$

## 3d. Let’s do another one using the methods we’ll be very familiar with by now

$-2( 4-x) =6( x+2) +3x$

Multiply out the brackets:

$-8+ 2x=6x+12+3x$

Separate out and collect the numbers and $x$ terms:

$2x-6x-3x=12+8$

Simplify:

$-7x=20$

$x=- frac{20}{7}$

## 3e. Try this one out

$4( x-3) -( x-5) =0$

Multiply out the brackets:

$4x-12-x+5=0$

Separate and collect our numbers and $x$ terms:

$3x-7=0$

Simplify:

$3x= 7$

Therefore:

$x=frac{7}{3}$

## 3f. Let’s have one last go. You should be a natural by now!

$6+3x=5( x-1) -3( x-2)$

Multiply out the brackets:

$6+3x=5x-5-3x+6$

Simplify:

$6+3x=2x+1$

Separate out and collect:

$3x-2x=1-6$

Therefore $x= -5$

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.

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