Simultaneous equations might look scary, but they're actually just puzzles... Show more
Maths599 views·Updated May 28, 2026·6 pagesUnderstanding Simultaneous Equations
Charlotte@charlotte_xx
1 / 6
1
of 6
Getting Started with Elimination
The elimination method is your best mate for solving simultaneous equations quickly. You're basically trying to make one of the variables disappear so you can solve for the other one.
Look at this example: when you subtract the second equation from the first, the 3y terms cancel out perfectly, leaving you with 2x = 8. This means x = 4, and you can substitute this back to find y = 2.
Quick Tip: Always check your answer by plugging both values back into the original equations - if they work, you've nailed it!
The key is spotting which variable will eliminate easily, then using that to your advantage.
2
of 6
Mastering the Substitution Back
Once you've found one variable using elimination, substituting back becomes dead simple. Take the first example: after finding x = 3, you just pop it into either original equation to find y.
With 5x + 4y = 43, substitute x = 3 to get 15 + 4y = 43, which gives you y = 7. The solution is (3,7).
Pro Tip: Choose the simpler-looking equation for substitution - it'll save you time and reduce calculation errors!
Practice makes perfect with these substitutions, so don't worry if it feels clunky at first.
3
of 6
When Coefficients Line Up Perfectly
Sometimes the maths gods smile on you, and the coefficients are already set up for easy elimination. In this example, you've got +4y and -4y, which cancel out beautifully when you add the equations.
Adding 7x + 4y = 58 and 5x - 4y = 14 gives you 12x = 72, so x = 6. Substitute back to find y = 4, and you're done!
Remember: Addition and subtraction both work for elimination - choose whichever makes the coefficients cancel out.
This is the dream scenario that makes simultaneous equations feel like a breeze.
4
of 6
Multiplying to Make It Work
When the coefficients don't naturally eliminate, you need to multiply one or both equations to create matching terms. This is where simultaneous equations get a bit more involved, but the process stays the same.
In the first example, multiplying the first equation by 2 and the second by 3 creates matching y coefficients (6y). Then you can subtract to eliminate y and solve for x.
Strategy Alert: Look for the lowest common multiple of the coefficients you want to eliminate - it keeps the numbers manageable.
The second example shows multiplying by larger numbers, but the principle is identical. Find x first, then substitute to get y.
5
of 6
Real-World Problems with Coffee and Cakes
Word problems with simultaneous equations pop up everywhere, especially in shops and pricing scenarios. The trick is translating the words into maths.
Let x = price of coffee and y = price of cake. "2 coffees and 3 cakes cost £9.95" becomes 2x + 3y = 9.95. "1 coffee and 4 cakes cost £10.35" becomes x + 4y = 10.35.
Translation Tip: Always define your variables clearly at the start - it prevents confusion later!
Solve these exactly like any other simultaneous equations, then remember to interpret your answer in context (coffee costs £2, cake costs £1.65).
6
of 6
Sweet Problems with Packs
This sweets problem shows how simultaneous equations handle real inventory situations. Let x = sweets in small pack and y = sweets in big pack.
"4 small packs and 3 big packs contain 175 sweets" gives you 4x + 3y = 175. "5 small packs and 2 big packs contain 154 sweets" gives you 5x + 2y = 154.
Real-World Reminder: Your answers should make sense - negative sweets or fractional packs usually mean you've made an error!
Using elimination (multiply first equation by 2, second by 3), you'll find that small packs contain 16 sweets and big packs contain 37 sweets.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1079,7556,318
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,33054
M
MathsMedium Level alerbra
Master challenging maths concepts with this medium level flashcard set designed for grade 7/8 students. Strengthen your problem-solving skills and boost your confidence in maths!
75533
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1221,9901,818
M
MathsMastering Maths: Essential Concepts for Grade 10
Boost your math skills with this comprehensive flashcard set covering key concepts for grade 10. Perfect for exam preparation and building a strong foundation in mathematics.
104261
M
MathsMastering Medium-Level Maths: Essential Flashcards for Grade 11 Students
Boost your Maths skills with this comprehensive set of flashcards designed specifically for Grade 11 students. Covering medium-level topics, these cards will help you ace your exams and build a solid foundation for advanced Maths.
118823
MathsComprehensive Maths Concepts
Explore essential higher mathematics concepts including calculus, trigonometry, polynomials, and vector analysis. This summary covers key topics such as differentiation, integration, quadratic equations, and the properties of circles, providing a solid foundation for exam preparation. Ideal for students seeking a concise yet thorough review of advanced mathematical principles.
S51,93757
P
MathsPercentage,fractions and decimals
how well do you know percentages,fractions and decimals
72703
M
Mathsmaths SOHCAHTOA
Trigonometric ratios SOHCAHTOA for calculating angles and sides in right-angled triangles.
111740
Most popular content
9
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12102,3893,038
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,8071,059
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,2392,304
English LiteratureAn Inspector Calls: Character Insights
Explore in-depth analysis and key quotes for characters in J.B. Priestley's 'An Inspector Calls'. This resource covers Gerald Croft, Inspector Goole, Sheila Birling, Mrs. Birling, Eric Birling, and Eva Smith, focusing on themes of class, gender roles, and social responsibility. Ideal for students aiming for Grade 8 and above.
1025,231901
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,119124
CriminologyCriminology Theories Overview
Explore key criminology theories and their implications on crime and deviance. This comprehensive summary covers biological, psychological, and sociological perspectives, including labelling theory, right realism, and the impact of social campaigns on policy development. Ideal for A-Level criminology students seeking to understand the complexities of criminal behaviour and the factors influencing crime prevention strategies.
129,746211
English LiteratureRomeo and Juliet: Key themes
Key Romeo and Juliet themes and analysed quotes
106,618197
C
BiologyCell Biology and Cell structure
cell structures
92,6560
English LiteratureMacbeth: Guilt and Ambition
Explore the complex themes of guilt and ambition in Shakespeare's 'Macbeth'. This analysis covers key characters, including Macbeth and Lady Macbeth, their moral dilemmas, and the tragic consequences of their ambition. Ideal for students studying character motivations, thematic elements, and the psychological impact of power. Includes insights on the natural order, manipulation, and the descent into madness.
918,818392
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user
Maths599 views·Updated May 28, 2026·6 pagesUnderstanding Simultaneous Equations
Charlotte@charlotte_xx
Simultaneous equations might look scary, but they're actually just puzzles where you find two unknown numbers that work in both equations. Once you master the elimination method, you'll be solving these confidently in no time!
1
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Getting Started with Elimination
The elimination method is your best mate for solving simultaneous equations quickly. You're basically trying to make one of the variables disappear so you can solve for the other one.
Look at this example: when you subtract the second equation from the first, the 3y terms cancel out perfectly, leaving you with 2x = 8. This means x = 4, and you can substitute this back to find y = 2.
Quick Tip: Always check your answer by plugging both values back into the original equations - if they work, you've nailed it!
The key is spotting which variable will eliminate easily, then using that to your advantage.
2
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Mastering the Substitution Back
Once you've found one variable using elimination, substituting back becomes dead simple. Take the first example: after finding x = 3, you just pop it into either original equation to find y.
With 5x + 4y = 43, substitute x = 3 to get 15 + 4y = 43, which gives you y = 7. The solution is (3,7).
Pro Tip: Choose the simpler-looking equation for substitution - it'll save you time and reduce calculation errors!
Practice makes perfect with these substitutions, so don't worry if it feels clunky at first.
3
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
When Coefficients Line Up Perfectly
Sometimes the maths gods smile on you, and the coefficients are already set up for easy elimination. In this example, you've got +4y and -4y, which cancel out beautifully when you add the equations.
Adding 7x + 4y = 58 and 5x - 4y = 14 gives you 12x = 72, so x = 6. Substitute back to find y = 4, and you're done!
Remember: Addition and subtraction both work for elimination - choose whichever makes the coefficients cancel out.
This is the dream scenario that makes simultaneous equations feel like a breeze.
4
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Multiplying to Make It Work
When the coefficients don't naturally eliminate, you need to multiply one or both equations to create matching terms. This is where simultaneous equations get a bit more involved, but the process stays the same.
In the first example, multiplying the first equation by 2 and the second by 3 creates matching y coefficients (6y). Then you can subtract to eliminate y and solve for x.
Strategy Alert: Look for the lowest common multiple of the coefficients you want to eliminate - it keeps the numbers manageable.
The second example shows multiplying by larger numbers, but the principle is identical. Find x first, then substitute to get y.
5
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Real-World Problems with Coffee and Cakes
Word problems with simultaneous equations pop up everywhere, especially in shops and pricing scenarios. The trick is translating the words into maths.
Let x = price of coffee and y = price of cake. "2 coffees and 3 cakes cost £9.95" becomes 2x + 3y = 9.95. "1 coffee and 4 cakes cost £10.35" becomes x + 4y = 10.35.
Translation Tip: Always define your variables clearly at the start - it prevents confusion later!
Solve these exactly like any other simultaneous equations, then remember to interpret your answer in context (coffee costs £2, cake costs £1.65).
6
of 6Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Sweet Problems with Packs
This sweets problem shows how simultaneous equations handle real inventory situations. Let x = sweets in small pack and y = sweets in big pack.
"4 small packs and 3 big packs contain 175 sweets" gives you 4x + 3y = 175. "5 small packs and 2 big packs contain 154 sweets" gives you 5x + 2y = 154.
Real-World Reminder: Your answers should make sense - negative sweets or fractional packs usually mean you've made an error!
Using elimination (multiply first equation by 2, second by 3), you'll find that small packs contain 16 sweets and big packs contain 37 sweets.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1079,7556,318
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,33054
M
MathsMedium Level alerbra
Master challenging maths concepts with this medium level flashcard set designed for grade 7/8 students. Strengthen your problem-solving skills and boost your confidence in maths!
75533
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1221,9901,818
M
MathsMastering Maths: Essential Concepts for Grade 10
Boost your math skills with this comprehensive flashcard set covering key concepts for grade 10. Perfect for exam preparation and building a strong foundation in mathematics.
104261
M
MathsMastering Medium-Level Maths: Essential Flashcards for Grade 11 Students
Boost your Maths skills with this comprehensive set of flashcards designed specifically for Grade 11 students. Covering medium-level topics, these cards will help you ace your exams and build a solid foundation for advanced Maths.
118823
MathsComprehensive Maths Concepts
Explore essential higher mathematics concepts including calculus, trigonometry, polynomials, and vector analysis. This summary covers key topics such as differentiation, integration, quadratic equations, and the properties of circles, providing a solid foundation for exam preparation. Ideal for students seeking a concise yet thorough review of advanced mathematical principles.
S51,93757
P
MathsPercentage,fractions and decimals
how well do you know percentages,fractions and decimals
72703
M
Mathsmaths SOHCAHTOA
Trigonometric ratios SOHCAHTOA for calculating angles and sides in right-angled triangles.
111740
Most popular content
9SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12102,3893,038
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,8071,059
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,2392,304
English LiteratureAn Inspector Calls: Character Insights
Explore in-depth analysis and key quotes for characters in J.B. Priestley's 'An Inspector Calls'. This resource covers Gerald Croft, Inspector Goole, Sheila Birling, Mrs. Birling, Eric Birling, and Eva Smith, focusing on themes of class, gender roles, and social responsibility. Ideal for students aiming for Grade 8 and above.
1025,231901
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,119124
CriminologyCriminology Theories Overview
Explore key criminology theories and their implications on crime and deviance. This comprehensive summary covers biological, psychological, and sociological perspectives, including labelling theory, right realism, and the impact of social campaigns on policy development. Ideal for A-Level criminology students seeking to understand the complexities of criminal behaviour and the factors influencing crime prevention strategies.
129,746211
English LiteratureRomeo and Juliet: Key themes
Key Romeo and Juliet themes and analysed quotes
106,618197
C
BiologyCell Biology and Cell structure
cell structures
92,6560
English LiteratureMacbeth: Guilt and Ambition
Explore the complex themes of guilt and ambition in Shakespeare's 'Macbeth'. This analysis covers key characters, including Macbeth and Lady Macbeth, their moral dilemmas, and the tragic consequences of their ambition. Ideal for students studying character motivations, thematic elements, and the psychological impact of power. Includes insights on the natural order, manipulation, and the descent into madness.
918,818392
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user