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Biology
Chemistry
AP Chemistry
English
AP English Literature
Geometry
Dec 8, 2025
150
7 pages
Bradley @himhimself
Algebraic proofs use mathematical properties to logically demonstrate why equations work the way they do. Instead of just... Show more
Algebraic proofs are structured ways to show your mathematical reasoning step-by-step. Unlike regular math problems where you just find the answer, proofs require you to explain why each step is valid.
Think of a proof as showing your work with explanations that anyone could follow and verify. This skill helps develop logical thinking that's useful far beyond math class.
You'll need to use specific properties and rules to justify every mathematical move you make, similar to citing evidence in an essay.
Remember Proofs aren't just about getting the right answer—they're about proving why that answer must be correct!
The properties of equality are the rules that let us manipulate equations while keeping them balanced. The addition property says if a = b, then a + c = b + c, while the subtraction property works similarly for taking away values.
For multiplication and division, we can multiply or divide both sides by the same number (except dividing by zero isn't allowed). The distributive property helps expand expressions.
Other key properties include the substitution property (replacing equal values), reflexive property , symmetric property , and transitive property .
Pro Tip These properties might seem obvious, but naming them in proofs shows you understand exactly why each step is mathematically valid.
When writing algebraic proofs, you'll typically use a two-column format that shows what you're doing and why you're allowed to do it. The left column contains mathematical statements, while the right column provides reasons for each step.
Your reasons will usually reference one of the properties of equality, like "addition property" or "distributive property." Start with what's given, then work through your reasoning steps, and finish with your conclusion.
This organized approach forces you to think about the "why" behind each algebra move instead of just rushing to the answer. Many students find this challenging at first but very rewarding once mastered.
Think about it In regular math problems, you just need the right answer. In proofs, the journey matters just as much as the destination!
Let's see how a simple proof works with the equation 4x - 1 = 27. Our goal is to prove that x = 7 using proper justification for each step.
In the first statement, we write what we know 4x - 1 = 27 (given). Next, we add 1 to both sides to get 4x = 28, citing the addition property of equality as our reason. Finally, we divide both sides by 4 to get x = 7, using the division property as justification.
This example shows how even simple equation-solving becomes a formal proof when we carefully justify each step with mathematical properties.
Quick Check Could you solve this equation in your head? Probably! But in a proof, showing your reasoning step-by-step is what matters.
When dealing with fractions in proofs, we apply the same properties but need to be extra careful with signs. In this problem, we start with a/(-6) + 2 = 5 and need to prove a = -18.
First, we subtract 2 from both sides using the subtraction property, getting a/(-6) = 3. Then, we multiply both sides by -6 using the multiplication property to find a = -18.
Notice how we're creating a logical chain where each step follows necessarily from the one before it. This is the power of mathematical proof—it removes any doubt about whether the answer is correct.
Remember When working with negative numbers in fractions, be especially careful tracking the signs as you apply properties!
This proof has more steps but follows the same logical pattern. Starting with 6x + 7 = 8x - 17, we need to prove x = 12.
We begin by subtracting 6x from both sides (subtraction property) to get 7 = 2x - 17. Then we add 17 to both sides (addition property) to find 24 = 2x. Dividing both sides by 2 (division property) gives us 12 = x. Finally, we use the symmetric property to write x = 12.
The final step using the symmetric property might seem unnecessary, but in formal proofs, even flipping the equation requires justification. This attention to detail is what makes proofs powerful.
Challenge yourself Try rewriting this proof without the symmetric property at the end. Would you need a different approach?
Here we tackle a more complex equation -7 + 4x = 6. This requires using the distributive property before we can solve for x = 2/3.
First, we expand both sides using the distributive property -7x - 14 + 4x = 12x - 24. Then we combine like terms on the left side -3x - 14 = 12x - 24. Subtracting 12x from both sides gives us -15x - 14 = -24.
Adding 14 to both sides simplifies to -15x = -10. Finally, dividing both sides by -15 produces our answer x = 2/3.
You've got this! Even complex proofs just break down into a series of simple steps. If you can justify each move, you can handle any algebraic proof!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Use the properties of equality to complete each proof
GeometryProperties of parallelogram notes
GeometryThis Polygons and Quadrilaterals Review discusses the Polygon Angle-Sum Theorems, polygon exterior Angle-Sum Theorem, Properties of Parallelograms, Properties of Rhombuses, Rectangles, and Squares, and Trapezoids and Kites.
GeometryCompleting the proofs below by giving the missing statements and reasons.
GeometryLearn about the properties of a rhombus, including congruent sides, perpendicular diagonals, and angle bisectors, with examples and equations.
GeometryMy filled out notes of these proofs and notes. There is an EMPTY version as well for practice.
GeometryApp Store
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Bradley
@himhimself
Algebraic proofs use mathematical properties to logically demonstrate why equations work the way they do. Instead of just solving equations, we use specific properties to justify each step, creating a formal proof that shows why our solution is correct and... Show more
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Join milions of students
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Algebraic proofs are structured ways to show your mathematical reasoning step-by-step. Unlike regular math problems where you just find the answer, proofs require you to explain why each step is valid.
Think of a proof as showing your work with explanations that anyone could follow and verify. This skill helps develop logical thinking that's useful far beyond math class.
You'll need to use specific properties and rules to justify every mathematical move you make, similar to citing evidence in an essay.
Remember: Proofs aren't just about getting the right answer—they're about proving why that answer must be correct!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The properties of equality are the rules that let us manipulate equations while keeping them balanced. The addition property says if a = b, then a + c = b + c, while the subtraction property works similarly for taking away values.
For multiplication and division, we can multiply or divide both sides by the same number (except dividing by zero isn't allowed). The distributive property helps expand expressions.
Other key properties include the substitution property (replacing equal values), reflexive property , symmetric property , and transitive property .
Pro Tip: These properties might seem obvious, but naming them in proofs shows you understand exactly why each step is mathematically valid.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When writing algebraic proofs, you'll typically use a two-column format that shows what you're doing and why you're allowed to do it. The left column contains mathematical statements, while the right column provides reasons for each step.
Your reasons will usually reference one of the properties of equality, like "addition property" or "distributive property." Start with what's given, then work through your reasoning steps, and finish with your conclusion.
This organized approach forces you to think about the "why" behind each algebra move instead of just rushing to the answer. Many students find this challenging at first but very rewarding once mastered.
Think about it: In regular math problems, you just need the right answer. In proofs, the journey matters just as much as the destination!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Let's see how a simple proof works with the equation 4x - 1 = 27. Our goal is to prove that x = 7 using proper justification for each step.
In the first statement, we write what we know: 4x - 1 = 27 (given). Next, we add 1 to both sides to get 4x = 28, citing the addition property of equality as our reason. Finally, we divide both sides by 4 to get x = 7, using the division property as justification.
This example shows how even simple equation-solving becomes a formal proof when we carefully justify each step with mathematical properties.
Quick Check: Could you solve this equation in your head? Probably! But in a proof, showing your reasoning step-by-step is what matters.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When dealing with fractions in proofs, we apply the same properties but need to be extra careful with signs. In this problem, we start with a/(-6) + 2 = 5 and need to prove a = -18.
First, we subtract 2 from both sides using the subtraction property, getting a/(-6) = 3. Then, we multiply both sides by -6 using the multiplication property to find a = -18.
Notice how we're creating a logical chain where each step follows necessarily from the one before it. This is the power of mathematical proof—it removes any doubt about whether the answer is correct.
Remember: When working with negative numbers in fractions, be especially careful tracking the signs as you apply properties!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This proof has more steps but follows the same logical pattern. Starting with 6x + 7 = 8x - 17, we need to prove x = 12.
We begin by subtracting 6x from both sides (subtraction property) to get 7 = 2x - 17. Then we add 17 to both sides (addition property) to find 24 = 2x. Dividing both sides by 2 (division property) gives us 12 = x. Finally, we use the symmetric property to write x = 12.
The final step using the symmetric property might seem unnecessary, but in formal proofs, even flipping the equation requires justification. This attention to detail is what makes proofs powerful.
Challenge yourself: Try rewriting this proof without the symmetric property at the end. Would you need a different approach?
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here we tackle a more complex equation: -7 + 4x = 6. This requires using the distributive property before we can solve for x = 2/3.
First, we expand both sides using the distributive property: -7x - 14 + 4x = 12x - 24. Then we combine like terms on the left side: -3x - 14 = 12x - 24. Subtracting 12x from both sides gives us -15x - 14 = -24.
Adding 14 to both sides simplifies to -15x = -10. Finally, dividing both sides by -15 produces our answer: x = 2/3.
You've got this! Even complex proofs just break down into a series of simple steps. If you can justify each move, you can handle any algebraic proof!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Use the properties of equality to complete each proof
GeometryProperties of parallelogram notes
GeometryThis Polygons and Quadrilaterals Review discusses the Polygon Angle-Sum Theorems, polygon exterior Angle-Sum Theorem, Properties of Parallelograms, Properties of Rhombuses, Rectangles, and Squares, and Trapezoids and Kites.
GeometryCompleting the proofs below by giving the missing statements and reasons.
GeometryLearn about the properties of a rhombus, including congruent sides, perpendicular diagonals, and angle bisectors, with examples and equations.
GeometryMy filled out notes of these proofs and notes. There is an EMPTY version as well for practice.
GeometryApp Store
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Calculus 1
Algebra 1
Overige