Linear equations are the building blocks of algebra, helping us... Show more
Math 8th Grade
Algebra 1
Pre-Calculus
Algebra 2
Calculus 1
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Linear Equations Basics
An equation is simply a statement that two quantities are equal. When you see something like 3x - 5 = 7, you're looking at an equation with two members (the left side and right side) and a variable (x) whose value you need to find. The value that makes the equation true is called the solution or root.
Solving linear equations (equations with variables raised only to the first power) follows simple principles. You can add, subtract, multiply, or divide both sides by the same number without changing the solution. The goal is to isolate the variable on one side.
To solve a linear equation:
- Move all terms with the variable to one side and all other terms to the other side
- Combine like terms to simplify
- Divide both sides by the coefficient of the variable
- Check your answer by substituting it back into the original equation
💡 When solving equations, think of it like keeping a balance scale equal. Whatever operation you perform on one side, you must do the same to the other side!
Some equations might give you extraneous roots - solutions that don't actually work in the original equation. Always check your answers to make sure they truly solve the problem.
Working with Fractions and Literal Equations
Linear equations often contain fractions, which can make them look intimidating. The secret is to multiply both sides by the least common multiple (LCM) of all the denominators to clear the fractions. This transforms the equation into a simpler form without changing its solution.
A literal equation contains multiple variables or letters, like the formula for interest I = PRT. When asked to solve for a specific variable, you use the same techniques as before, but your goal is to isolate that particular letter. For example, to solve for R in I = PRT, you would divide both sides by PT to get R = I/(PT).
Literal equations are incredibly practical since they show up in science, engineering, and finance. Learning to rearrange these formulas helps you adapt them to different situations without memorizing multiple versions of the same relationship.
🌟 Remember that literal equations are just regular equations where some constants are represented by letters instead of specific numbers!
When solving for a specific variable, treat all other variables as if they were numbers, moving them around using the same rules you'd use for solving regular equations.
Systems of Linear Equations
Sometimes one equation isn't enough to find all the unknowns. A system of linear equations consists of two or more equations that need to be solved simultaneously. For two variables (x and y), we need at least two equations to find a unique solution.
There are three main methods to solve systems of two linear equations:
-
Graphical Method: Plot both equations on a graph. The intersection point gives the solution. This method provides a visual understanding but may lack precision.
-
Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation. This works well when one equation has a variable with a coefficient of 1.
-
Elimination Method: Manipulate the equations so that one variable has coefficients that are opposites, then add the equations to eliminate that variable. This is particularly useful when coefficients are similar in magnitude.
🔍 The elimination method is usually fastest when both equations have similar coefficients, while substitution often works better when one variable can be easily isolated!
For systems with fractions, start by clearing all fractions before applying any method. This makes the calculations much cleaner and helps avoid arithmetic errors.
Advanced Systems and Word Problems
Systems of three linear equations require finding values for three variables (typically x, y, and z). The elimination method works by eliminating the same variable from different pairs of equations, creating a system of two equations that you already know how to solve.
Cramer's Rule offers an elegant approach using determinants to solve systems of equations. While it may seem complex initially, it provides a systematic method that works for any number of equations and variables without requiring substitutions.
Tackling word problems requires a structured approach:
- Read the entire problem quickly to understand what type of problem it is
- Identify what you're solving for (often stated in the question at the end)
- Let x (or another variable) represent the unknown quantity
- If there are multiple unknowns, express them in terms of your primary variable
- Translate the problem conditions into an equation or system of equations
💪 The hardest part of word problems isn't the math—it's translating from English to algebra! Practice identifying which phrases represent equality, addition, multiplication, etc.
When setting up your equation, be sure to clearly label what each variable represents. This not only helps you solve the problem correctly but also makes it easier to verify your answer makes sense in the original context.
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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
Introduction to Linear Equations
Linear equations are the building blocks of algebra, helping us find unknown values by creating mathematical relationships. Whether solving for one variable or working with multiple equations, the skills you'll learn here are essential for tackling both everyday problems and... Show more
Linear Equations Basics
An equation is simply a statement that two quantities are equal. When you see something like 3x - 5 = 7, you're looking at an equation with two members (the left side and right side) and a variable (x) whose value you need to find. The value that makes the equation true is called the solution or root.
Solving linear equations (equations with variables raised only to the first power) follows simple principles. You can add, subtract, multiply, or divide both sides by the same number without changing the solution. The goal is to isolate the variable on one side.
To solve a linear equation:
- Move all terms with the variable to one side and all other terms to the other side
- Combine like terms to simplify
- Divide both sides by the coefficient of the variable
- Check your answer by substituting it back into the original equation
💡 When solving equations, think of it like keeping a balance scale equal. Whatever operation you perform on one side, you must do the same to the other side!
Some equations might give you extraneous roots - solutions that don't actually work in the original equation. Always check your answers to make sure they truly solve the problem.
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Working with Fractions and Literal Equations
Linear equations often contain fractions, which can make them look intimidating. The secret is to multiply both sides by the least common multiple (LCM) of all the denominators to clear the fractions. This transforms the equation into a simpler form without changing its solution.
A literal equation contains multiple variables or letters, like the formula for interest I = PRT. When asked to solve for a specific variable, you use the same techniques as before, but your goal is to isolate that particular letter. For example, to solve for R in I = PRT, you would divide both sides by PT to get R = I/(PT).
Literal equations are incredibly practical since they show up in science, engineering, and finance. Learning to rearrange these formulas helps you adapt them to different situations without memorizing multiple versions of the same relationship.
🌟 Remember that literal equations are just regular equations where some constants are represented by letters instead of specific numbers!
When solving for a specific variable, treat all other variables as if they were numbers, moving them around using the same rules you'd use for solving regular equations.
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Systems of Linear Equations
Sometimes one equation isn't enough to find all the unknowns. A system of linear equations consists of two or more equations that need to be solved simultaneously. For two variables (x and y), we need at least two equations to find a unique solution.
There are three main methods to solve systems of two linear equations:
-
Graphical Method: Plot both equations on a graph. The intersection point gives the solution. This method provides a visual understanding but may lack precision.
-
Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation. This works well when one equation has a variable with a coefficient of 1.
-
Elimination Method: Manipulate the equations so that one variable has coefficients that are opposites, then add the equations to eliminate that variable. This is particularly useful when coefficients are similar in magnitude.
🔍 The elimination method is usually fastest when both equations have similar coefficients, while substitution often works better when one variable can be easily isolated!
For systems with fractions, start by clearing all fractions before applying any method. This makes the calculations much cleaner and helps avoid arithmetic errors.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Systems and Word Problems
Systems of three linear equations require finding values for three variables (typically x, y, and z). The elimination method works by eliminating the same variable from different pairs of equations, creating a system of two equations that you already know how to solve.
Cramer's Rule offers an elegant approach using determinants to solve systems of equations. While it may seem complex initially, it provides a systematic method that works for any number of equations and variables without requiring substitutions.
Tackling word problems requires a structured approach:
- Read the entire problem quickly to understand what type of problem it is
- Identify what you're solving for (often stated in the question at the end)
- Let x (or another variable) represent the unknown quantity
- If there are multiple unknowns, express them in terms of your primary variable
- Translate the problem conditions into an equation or system of equations
💪 The hardest part of word problems isn't the math—it's translating from English to algebra! Practice identifying which phrases represent equality, addition, multiplication, etc.
When setting up your equation, be sure to clearly label what each variable represents. This not only helps you solve the problem correctly but also makes it easier to verify your answer makes sense in the original context.
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