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Paylee
11/17/2025
Geometry
Algebraic Proofs
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Algebraic Proofs: Transforming Equations
Algebraic proofs use properties of equality to show how one equation can be derived from another. Each step in a proof needs both a statement and a reason that justifies the change.
Let's look at some examples. When proving that "y = 3x - 7" from "6x - 2y = 14," we start by isolating the y-term using subtraction, getting "-2y = -6x + 14." Then we divide both sides by -2 to find our answer.
Similarly, when working with more complex proofs like "b = 2m - a" from "m = ½," we multiply both sides by 2 first, then rearrange using subtraction. The substitution property is especially useful when we have multiple given equations, allowing us to replace equal expressions.
Remember: Each step in your proof needs both a clear statement (what you did) and a valid reason (why you were allowed to do it).
When solving multi-step proofs like "q = 5" from "p = 3q, p + q = r, r = 20," you'll need to carefully substitute one equation into another until you reach your goal. Always think about which property will help simplify your expression next.
Working Through More Complex Proofs
Ever wondered how to tackle more challenging algebraic proofs? Let's see how to prove "y = -2" when given "2x + y = 4" and "x = y + 5." The key is using substitution effectively to combine information from both equations.
By replacing x with in the first equation, we get "2 + y = 4." Then we distribute, combine like terms, and solve for y. This methodical approach works for any algebraic proof.
Temperature conversion formulas make great examples of algebraic proofs. When proving "C = 9/5" from "F = 5/9C + 32," we isolate C using subtraction and multiplication properties. Each step brings us closer to our goal.
Pro tip: When working with multiple equations, look for opportunities to create equal expressions that can be compared using the transitive property of equality.
The proof "x = v" from "x + y = z, w + v = z, w = y" shows how powerful substitution can be. By recognizing that both expressions equal z, we can set them equal to each other. Then by substituting w = y, we can simplify to our target statement through careful algebraic manipulation.
Practicing with Different Equation Types
Ready to level up your algebraic proof skills? Let's tackle some homework problems with different equation types. For example, to prove "c = 13d" from "4d = 1/3," we start by multiplying both sides by 3 to eliminate the fraction.
After getting "12d = c - d," we use the addition property of equality to move all terms with d to one side, resulting in "13d = c." The final step simply uses the symmetric property to write "c = 13d."
Fractions require special attention in proofs. When proving "j = 10 - 3k" from "/2 = 15 - 4k," we first multiply both sides by 2 to simplify. Then we isolate the j-term through careful algebra.
Think of it this way: Each proof is like solving a puzzle where you need to find the right sequence of moves to reach your destination.
Working with multiple equations means looking for smart substitutions. In problems like proving "m = 3a" from "a + m = n" and "n = 4a," we substitute the second equation into the first. This gives us "a + m = 4a," and with one more step, we reach our goal of "m = 3a."
Mastering Multi-Step Proofs
Multi-step proofs often involve several given equations that need to be connected strategically. To prove "p = s" from "p + q = 25," "r + s = 25," and "q = r," we first recognize that the first two equations show "p + q = r + s."
By substituting "q = r" into this equation, we get "p + r = r + s." A simple subtraction of r from both sides leads us to "p = s." The substitution property is your best friend in these complex proofs!
When variables appear in different forms, create a pathway to your goal. For example, proving "f = 3" from "2e + 10 = 16 - 2h" and "e = f - h" requires careful substitution and algebraic manipulation. Replace e with , distribute, and then simplify.
Challenge yourself: Can you predict which properties you'll need before starting a proof? This skill helps you plan your approach more efficiently.
The most complex proofs might involve systems of equations. To prove "b = -1" from "a - 2b = 8" and "4a + 3b = 21," first solve for a in terms of b. Then substitute this expression into the second equation. After distributing and combining like terms, you'll find that b equals -1. Each step builds logically on the previous one until you reach your destination.
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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
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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
Mastering Algebraic Proofs Using Equality Properties
Algebraic proofs show how we can use properties of equality to prove mathematical statements. By applying these properties step-by-step, we can transform equations logically and demonstrate that one mathematical statement follows from another.
Algebraic Proofs: Transforming Equations
Algebraic proofs use properties of equality to show how one equation can be derived from another. Each step in a proof needs both a statement and a reason that justifies the change.
Let's look at some examples. When proving that "y = 3x - 7" from "6x - 2y = 14," we start by isolating the y-term using subtraction, getting "-2y = -6x + 14." Then we divide both sides by -2 to find our answer.
Similarly, when working with more complex proofs like "b = 2m - a" from "m = ½," we multiply both sides by 2 first, then rearrange using subtraction. The substitution property is especially useful when we have multiple given equations, allowing us to replace equal expressions.
Remember: Each step in your proof needs both a clear statement (what you did) and a valid reason (why you were allowed to do it).
When solving multi-step proofs like "q = 5" from "p = 3q, p + q = r, r = 20," you'll need to carefully substitute one equation into another until you reach your goal. Always think about which property will help simplify your expression next.
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Working Through More Complex Proofs
Ever wondered how to tackle more challenging algebraic proofs? Let's see how to prove "y = -2" when given "2x + y = 4" and "x = y + 5." The key is using substitution effectively to combine information from both equations.
By replacing x with in the first equation, we get "2 + y = 4." Then we distribute, combine like terms, and solve for y. This methodical approach works for any algebraic proof.
Temperature conversion formulas make great examples of algebraic proofs. When proving "C = 9/5" from "F = 5/9C + 32," we isolate C using subtraction and multiplication properties. Each step brings us closer to our goal.
Pro tip: When working with multiple equations, look for opportunities to create equal expressions that can be compared using the transitive property of equality.
The proof "x = v" from "x + y = z, w + v = z, w = y" shows how powerful substitution can be. By recognizing that both expressions equal z, we can set them equal to each other. Then by substituting w = y, we can simplify to our target statement through careful algebraic manipulation.
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Practicing with Different Equation Types
Ready to level up your algebraic proof skills? Let's tackle some homework problems with different equation types. For example, to prove "c = 13d" from "4d = 1/3," we start by multiplying both sides by 3 to eliminate the fraction.
After getting "12d = c - d," we use the addition property of equality to move all terms with d to one side, resulting in "13d = c." The final step simply uses the symmetric property to write "c = 13d."
Fractions require special attention in proofs. When proving "j = 10 - 3k" from "/2 = 15 - 4k," we first multiply both sides by 2 to simplify. Then we isolate the j-term through careful algebra.
Think of it this way: Each proof is like solving a puzzle where you need to find the right sequence of moves to reach your destination.
Working with multiple equations means looking for smart substitutions. In problems like proving "m = 3a" from "a + m = n" and "n = 4a," we substitute the second equation into the first. This gives us "a + m = 4a," and with one more step, we reach our goal of "m = 3a."
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Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Mastering Multi-Step Proofs
Multi-step proofs often involve several given equations that need to be connected strategically. To prove "p = s" from "p + q = 25," "r + s = 25," and "q = r," we first recognize that the first two equations show "p + q = r + s."
By substituting "q = r" into this equation, we get "p + r = r + s." A simple subtraction of r from both sides leads us to "p = s." The substitution property is your best friend in these complex proofs!
When variables appear in different forms, create a pathway to your goal. For example, proving "f = 3" from "2e + 10 = 16 - 2h" and "e = f - h" requires careful substitution and algebraic manipulation. Replace e with , distribute, and then simplify.
Challenge yourself: Can you predict which properties you'll need before starting a proof? This skill helps you plan your approach more efficiently.
The most complex proofs might involve systems of equations. To prove "b = -1" from "a - 2b = 8" and "4a + 3b = 21," first solve for a in terms of b. Then substitute this expression into the second equation. After distributing and combining like terms, you'll find that b equals -1. Each step builds logically on the previous one until you reach your destination.
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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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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