Simultaneous equations might look scary, but they're actually just puzzles... Show more
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Updated Mar 26, 2026
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Understanding Simultaneous Equations
Charlotte
@charlotte_xx
1 / 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.
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.
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.
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.
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).
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.
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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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding 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!
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.
Sign up to see the contentIt's free!
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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.
Sign up to see the contentIt'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.
Sign up to see the contentIt'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.
Sign up to see the contentIt'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).
Sign up to see the contentIt'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.
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Most popular content in Maths
maths helpbook to get grade 9
material from online, not my personal advice
Maths9
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
Google 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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user