Simultaneous equations are a common topic in National 5 Maths and regularly appear in assessments and exams. While they can seem challenging at first, they become much easier once you understand the methods involved.
In this guide, we’ll explain what simultaneous equations are, how to solve them using substitution and elimination, and work through several National 5-style examples step by step.
By the end, you’ll have the confidence to tackle simultaneous equations in your own maths exams.
If you are looking for a National 5 maths tutor in Scotland, get in touch with Central Tutors today, we have a wide range of maths (and other subject) tutors available for both private and group tutoring.
What Are Simultaneous Equations?
Simultaneous equations are two equations that contain the same variables.
For example:
x + y = 10
x – y = 2
The aim is to find values for both x and y that satisfy both equations at the same time.
These values are known as the solution to the simultaneous equations. Read more aobut simultaneous equations on the BBC bitesize website.
Why Are Simultaneous Equations Important in National 5 Maths?
Simultaneous equations appear regularly in National 5 Maths because they test your ability to:
- Solve algebraic problems
- Rearrange equations
- Use logical reasoning
- Apply mathematical methods accurately
Understanding simultaneous equations also provides a foundation for Higher Maths and many STEM-related subjects.
You may find some of our other blog posts useful:
Simultaneous Equations Made Easy: Tips for National 5 Students in Scotland
Simultaneous Equations – what they are and how to use them
Method 1: Solving Simultaneous Equations Using Elimination
The elimination method involves removing one variable so that you can solve for the other.
Example
x + y = 10
x – y = 2
Step 1: Add the Equations Together
(x + y) + (x – y) = 10 + 2
2x = 12
Step 2: Solve for x
x = 6
Step 3: Substitute Back Into One Equation
6 + y = 10
y = 4
Answer
x = 6
y = 4
Method 2: Solving Simultaneous Equations Using Substitution
The substitution method works by expressing one variable in terms of another.
Example
y = 2x + 1
x + y = 10
Step 1: Substitute
Replace y in the second equation:
x + (2x + 1) = 10
Step 2: Simplify
3x + 1 = 10
3x = 9
x = 3
Step 3: Find y
y = 2(3) + 1
y = 7
Answer
x = 3
y = 7
Have a look at this other website to get another explanation, https://b28mathstutor.co.uk/solving-simultaneous-equations-part-2/.
National 5 Example Question
Here’s the BBC Bitesize page for National 5 Simultaneous Equations.
Solve:
2x + y = 13
x – y = 2
Step 1: Add the Equations
(2x + y) + (x – y) = 13 + 2
3x = 15
x = 5
Step 2: Substitute Back
5 – y = 2
y = 3
Answer
x = 5
y = 3
Simultaneous Equations in Word Problems
National 5 exams often include worded questions.
Example
A cinema sold 100 tickets.
Adult tickets cost £8.
Child tickets cost £5.
The total revenue was £650.
How many adult and child tickets were sold?
Let:
a = adult tickets
c = child tickets
Create two equations:
a + c = 100
8a + 5c = 650
These can then be solved using elimination.
Solving Simultaneous Equations Graphically
Simultaneous equations can be solved algebraically or graphically. To solve them graphically, both equations must be plotted on the same set of axes.
The solution is found at the point where the two graphs intersect. This point gives the values of x and y that satisfy both equations.
For example, consider the equations:
y = x + 1
x + y = 3
First, rearrange the second equation into the form y = mx + c:
y = 3 – x
The two graphs can then be plotted on the same set of axes.
When plotted, the graphs intersect at the point (1, 2).
This means the solution is:
x = 1
y = 2
You can check the solution by substituting the values into both equations:
y = x + 1
2 = 1 + 1 ✓
x + y = 3
1 + 2 = 3 ✓
Since both equations are satisfied, (1, 2) is the solution to the simultaneous equations.
Exam Tip
When solving simultaneous equations graphically, draw your lines carefully using a ruler and read the coordinates of the intersection as accurately as possible. In National 5 Maths exams, your answer may only be accepted if it is within an acceptable tolerance of the correct value.
The Ultimate Guide to Passing National 5 Maths in Scotland (2026 Revision Guide)
National 5 Maths Past Papers: Free SQA Exam Papers & Revision Resources
Common Mistakes Students Make
Forgetting to Eliminate Correctly
Always check that one variable cancels completely.
Sign Errors
Take care with positive and negative numbers when adding or subtracting equations.
Not Substituting Back
Once you find one variable, remember to calculate the second.
Poor Working Out
Examiners award marks for method as well as the final answer.
Show every step clearly.
National 5 Simultaneous Equations Practice Questions
Question 1
x + y = 12
x – y = 4
Question 2
2x + y = 11
x – y = 1
Question 3
3x + 2y = 16
x + y = 6
Question 4
y = x + 4
x + y = 14
Question 5
2x + 3y = 17
x + y = 7
Answers
Question 1
x = 8
y = 4
Question 2
x = 4
y = 3
Question 3
x = 4
y = 2
Question 4
x = 5
y = 9
Question 5
x = 4
y = 3
Revision Tips for Simultaneous Equations
- Practise both elimination and substitution methods.
- Show all working clearly.
- Check your answers by substituting them back into both equations.
- Complete past National 5 Maths papers.
- Focus on accuracy rather than speed.
The more simultaneous equations you solve, the easier they become.
Frequently Asked Questions
What are simultaneous equations?
Simultaneous equations are two or more equations containing the same variables that must be solved together.
Which method is best?
For most National 5 questions, elimination is usually the quickest method, although substitution is often useful when one equation is already rearranged.
Do simultaneous equations appear in National 5 Maths exams?
Yes. Simultaneous equations are a common National 5 Maths topic and students should be comfortable solving them using both elimination and substitution.
How can I improve at simultaneous equations?
Regular practice, checking your answers, and working through past-paper questions are the best ways to improve.
Simultaneous Equations: Key Takeaways
Simultaneous equations are an important topic in National 5 Maths and appear regularly in exams. Whether you solve them using elimination, substitution, or graphical methods, the key is to practise each technique until you feel confident applying it to different types of questions.
Remember that the solution to simultaneous equations is the pair of values that satisfy both equations at the same time. By working through examples, checking your answers, and completing past-paper questions, you’ll quickly improve your algebra skills and be better prepared for your National 5 Maths exam.
If you’re struggling with simultaneous equations or any other National 5 Maths topic, Central Tutors provides expert one-to-one tuition across Scotland. Our experienced tutors can help you build confidence, improve your understanding, and achieve the grades you’re aiming for.
Looking for extra support with National 5 Maths? Contact Central Tutors today to find out how our personalised tuition can help you succeed.
