Learning to solve three simultaneous equations might seem daunting, but...
Further Maths176 views·Updated 7 Jul 2026·1 pageSolving Three Simultaneous Equations - AQA Further Maths Revision
1
of 1
Solving Three Simultaneous Equations
Simultaneous equations with three unknowns require a strategic approach, but once you master the technique, you'll find it's quite straightforward. The key is systematically eliminating variables one by one.
Let's work through this example: x + y + z = 6 ①, 2x - y + z = 3 ②, and 3x + y - 2z = -1 ③. Start by adding equations to eliminate one variable - here we'll get rid of y first.
Adding equations ① and ② gives us 3x + 2z = 9 ④. Then adding equations ② and ③ gives us 5x - z = 2 ⑤. Notice how we now have two equations with just x and z!
To solve these remaining equations, multiply equation ⑤ by 2 to get 10x - 2z = 4 ⑥. Adding equations ④ and ⑥ eliminates z completely: 13x = 13, so x = 1. Substitute back to find z = 3, then use either original equation to find y = 2.
Top Tip: Always check your answer by substituting all three values back into each original equation - they should all balance perfectly!
We thought you’d never ask...
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Further Maths
9Further MathsAqa gcse level 2 further maths notes
These aren’t notes on all the topics! But most of them are there :)
1138210
Further MathsComplex Numbers & Matrices
Explore the fundamentals of complex numbers and matrices in this comprehensive study note. Topics include the square root of complex numbers, matrix transformations, determinants, and the properties of complex conjugates. Ideal for A Level Further Maths students looking to strengthen their understanding of these key concepts.
124,950134
Further MathsBivariate Data Analysis
Explore key concepts in bivariate data analysis, including hypothesis testing, correlation coefficients, and regression models. This summary covers significance levels, critical values, and the relationship between two variables, providing essential insights for WJEC AS Further Statistics. Ideal for students preparing for exams or seeking to understand statistical relationships.
12430
MathsProof Techniques Explained
Explore the four essential methods of proof in mathematics: deduction, exhaustion, counterexample, and contradiction. This summary provides clear definitions and examples for each technique, making it ideal for A-level students seeking to strengthen their understanding of mathematical proofs.
122292
MathsTrigonometry Topic Summary for AQA Further Maths
AQA Further Maths / GCSE maths. key information for the trig topic including trig identities and trig graphs
114004
Further MathsMatrix Operations & Transformations
Explore key concepts in matrix operations, including multiplication, determinants, inverses, and their applications in solving simultaneous equations and transformations. This summary covers essential properties and examples relevant for A-Level Further Mathematics (OCR/MEI).
1213911
Further MathsCircle Geometry Essentials
Explore key concepts of circle geometry, including the standard and expanded equations of circles, tangent lines, and the relationship between circle centers and points. This summary is ideal for GCSE Maths students looking to master circle theorems and equations.
101331
Further MathsUnderstanding Complex Numbers
Explore the fundamentals of complex numbers, including their real and imaginary parts, modulus, argument, and operations such as addition and multiplication. This summary covers key concepts like complex conjugates and polar form, essential for mastering WJEC AS Further Pure mathematics.
121011
MathsA-Level Maths 2022 Solutions
Explore detailed worked solutions for the A-Level Maths Paper 1 (2022). This resource covers key concepts such as differentiation, exponential functions, trigonometric equations, and area calculations, providing step-by-step explanations for each question. Ideal for students preparing for exams.
121,61958
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.
12102,9263,041
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.
1273,7182,307
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.
1254,8881,060
SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,6681,400
C
BiologyCell Biology and Cell structure
cell structures
93,2690
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,169125
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.
1025,455907
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.
129,769210
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,474846
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
Further Maths176 views·Updated 7 Jul 2026·1 pageSolving Three Simultaneous Equations - AQA Further Maths Revision
Learning to solve three simultaneous equations might seem daunting, but it's actually a systematic process that builds on what you already know about solving pairs of equations. This mathematical skill is essential for A-level maths and opens doors to understanding...
1
of 1
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Solving Three Simultaneous Equations
Simultaneous equations with three unknowns require a strategic approach, but once you master the technique, you'll find it's quite straightforward. The key is systematically eliminating variables one by one.
Let's work through this example: x + y + z = 6 ①, 2x - y + z = 3 ②, and 3x + y - 2z = -1 ③. Start by adding equations to eliminate one variable - here we'll get rid of y first.
Adding equations ① and ② gives us 3x + 2z = 9 ④. Then adding equations ② and ③ gives us 5x - z = 2 ⑤. Notice how we now have two equations with just x and z!
To solve these remaining equations, multiply equation ⑤ by 2 to get 10x - 2z = 4 ⑥. Adding equations ④ and ⑥ eliminates z completely: 13x = 13, so x = 1. Substitute back to find z = 3, then use either original equation to find y = 2.
Top Tip: Always check your answer by substituting all three values back into each original equation - they should all balance perfectly!
We thought you’d never ask...
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Further Maths
9Further MathsAqa gcse level 2 further maths notes
These aren’t notes on all the topics! But most of them are there :)
1138210
Further MathsComplex Numbers & Matrices
Explore the fundamentals of complex numbers and matrices in this comprehensive study note. Topics include the square root of complex numbers, matrix transformations, determinants, and the properties of complex conjugates. Ideal for A Level Further Maths students looking to strengthen their understanding of these key concepts.
124,950134
Further MathsBivariate Data Analysis
Explore key concepts in bivariate data analysis, including hypothesis testing, correlation coefficients, and regression models. This summary covers significance levels, critical values, and the relationship between two variables, providing essential insights for WJEC AS Further Statistics. Ideal for students preparing for exams or seeking to understand statistical relationships.
12430
MathsProof Techniques Explained
Explore the four essential methods of proof in mathematics: deduction, exhaustion, counterexample, and contradiction. This summary provides clear definitions and examples for each technique, making it ideal for A-level students seeking to strengthen their understanding of mathematical proofs.
122292
MathsTrigonometry Topic Summary for AQA Further Maths
AQA Further Maths / GCSE maths. key information for the trig topic including trig identities and trig graphs
114004
Further MathsMatrix Operations & Transformations
Explore key concepts in matrix operations, including multiplication, determinants, inverses, and their applications in solving simultaneous equations and transformations. This summary covers essential properties and examples relevant for A-Level Further Mathematics (OCR/MEI).
1213911
Further MathsCircle Geometry Essentials
Explore key concepts of circle geometry, including the standard and expanded equations of circles, tangent lines, and the relationship between circle centers and points. This summary is ideal for GCSE Maths students looking to master circle theorems and equations.
101331
Further MathsUnderstanding Complex Numbers
Explore the fundamentals of complex numbers, including their real and imaginary parts, modulus, argument, and operations such as addition and multiplication. This summary covers key concepts like complex conjugates and polar form, essential for mastering WJEC AS Further Pure mathematics.
121011
MathsA-Level Maths 2022 Solutions
Explore detailed worked solutions for the A-Level Maths Paper 1 (2022). This resource covers key concepts such as differentiation, exponential functions, trigonometric equations, and area calculations, providing step-by-step explanations for each question. Ideal for students preparing for exams.
121,61958
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.
12102,9263,041
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.
1273,7182,307
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.
1254,8881,060
SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,6681,400
C
BiologyCell Biology and Cell structure
cell structures
93,2690
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,169125
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.
1025,455907
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.
129,769210
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,474846
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