Simultaneous equations

Simultaneous equations form a crucial topic in the algebra section of the GCSE Maths syllabus. As algebra makes up the largest proportion of the course, it is essential that students develop confidence and accuracy when solving these types of problems.

At GCSE level, simultaneous equations are used to test a student’s ability to manipulate algebraic expressions, apply logical steps, and present clear working. These skills are not only important for exam success, but also form the foundation for further study at A-Level and beyond.

What are simultaneous equations?

Simultaneous equations are two (or more) equations that must be satisfied at the same time. In GCSE Maths, this usually means finding the values of two unknowns, such as x and y, that make both equations true.

For example:

The solution is the pair of values for x and y that satisfy both equations simultaneously.

Methods for solving simultaneous equations

At GCSE level, there are two main methods you are expected to know:

1. Elimination

This method involves adding or subtracting the equations so that one variable is eliminated, allowing you to solve for the other.

2. Substitution

This method involves rearranging one equation to make one variable the subject, then substituting this expression into the other equation.

Both methods are equally valid. The best approach depends on the structure of the equations you are given.

Example

Let’s attempt to solve the simultaneous equations presented previously. It would be worth labelling each equation (1) and (2) to make it easier to keep track of your workings.

Step 1: Choose a method

In this pair of equations, the coefficients of y are –1 and +1.

This makes the elimination method the most efficient choice, as the y-terms will cancel when the equations are added.

Step 2: Eliminate one variable

Add the two equations together:

(4x - y) + (6x + y) = 8 + 22

Simplifying:

10x = 30

Step 3: Solve for the remaining variable

Divide both sides by 10:

x = 3

Step 4: Substitute to find the second variable

Substitute x = 3 into either original equation.

Using 4x - y = 8:

4(3) - y = 8

12 - y = 8

y = 4

Step 5: Write the final answer

The solution to the simultaneous equations is:

x = 3, y = 4

Exam tips for simultaneous equations

• Always write down the method you are using clearly

• Show every algebraic step — method marks matter

• Check your final answers by substituting back into both equations

• If coefficients are awkward, consider multiplying one or both equations first

Clear presentation can make the difference between full marks and dropped marks.

Watch the full worked example

In the accompanying video on the math squared YouTube, we work through a GCSE simultaneous equations question step by step, explaining the method carefully and highlighting common mistakes to avoid.

Check out @mathsquaredorg on YouTube for more!

Good luck with your studies!

Will