Algebra 1

Dec 3, 2025

4 pages

Mastering Linear Equations: Forms and Solutions

Knowunity Official @team.knowunity

Linear relations and functions form the backbone of algebra, allowing us to model relationships between variables. In this... Show more

2.2 Linear Relations and Functions.
Linear Equations
-3x+5=Y
X-7=8
==x-3y=2
Non linear
y=x²-8
xy =3 ty
√3x-y="
2+y=8
Standard form: Ax + By

Linear Equations and Their Forms

Linear equations have variables with powers of 1, while non-linear equations have variables with other powers or are multiplied together. For example, "-3x+5=y" is linear, but "y=x²-8" is non-linear because x is squared.

There are several important forms of linear equations you need to know

Standard Form Ax + By = C (where A, B, C are integers and A ≥ 0)
Point-Slope Form y - y₁ = m $x - x₁$ (where m is the slope and (x₁,y₁) is a point on the line)
Slope-Intercept Form y = mx + b $where m is the slope and b is the y-intercept$

The x-intercept is where the line crosses the x-axis (also called zeros, roots, or solutions), while the y-intercept is where it crosses the y-axis. You can identify if an equation is linear by checking if all variables have power of 1 and aren't multiplied together.

💡 When trying to determine if an equation is linear, look for variables with exponents other than 1 or variables multiplied together (like xy) - these make the equation non-linear!

Working with Linear Equations

Let's analyze the equation 3x - y = 8

In standard form, we identify that A = 3, B = -1, and C = 8. Finding intercepts helps us graph the equation and understand its behavior.

To find the x-intercept, set y = 0 and solve for x 3x - 0 = 8 3x = 8 x = 8/3 So the x-intercept is (8/3, 0)

For the y-intercept, set x = 0 and solve for y 3(0) - y = 8 -y = 8 y = -8 So the y-intercept is (0, -8)

⚠️ Be careful with signs when rearranging equations! A common mistake is forgetting to change signs when moving terms from one side to another.

Converting Between Forms

When working with fractions in linear equations, you can multiply all terms by a common denominator to eliminate fractions. This makes the equation easier to work with.

For example, with ¼x + ⅘y = 4, we can multiply all terms by 20 (the LCM of 4 and 5) 5x + 16y = 80

Now we can easily identify the standard form values A = 5, B = 16, C = 80.

Finding intercepts follows the same process as before

For the x-intercept set y = 0, which gives us 5x = 80, so x = 16, making the point (16, 0)
For the y-intercept set x = 0, which gives us 16y = 80, so y = 5, making the point (0, 5)

🌟 Converting equations to standard form makes them easier to graph and analyze, especially when dealing with fractions!

Converting to Standard Form

Converting different equation types to standard form follows specific steps depending on the original form.

For equations like x - 3 = 9, first rearrange to isolate the variable x - 3 = 9 x = 12 This is now in the form x = 12, which technically isn't standard form, but tells us the line is vertical at x = 12.

For equations with decimals like -0.15x - 0.72 = 19y, multiply by 100 to eliminate decimals -15x - 72 = 19y Then rearrange to standard form 15x + 19y = -72

🔑 When converting to standard form, remember that A should be positive. If you get a negative A value, multiply the entire equation by -1 to fix it!

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.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam

Quiz

Flashcards

Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

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

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

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

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

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

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

Elisha

iOS user

Paul T

iOS user

Algebra 1

•

Dec 3, 2025

•

4 pages

Mastering Linear Equations: Forms and Solutions

Knowunity Official

@team.knowunity

Linear relations and functions form the backbone of algebra, allowing us to model relationships between variables. In this section, we'll explore different forms of linear equations, how to identify them, and techniques for finding key points on their graphs.

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

Linear Equations and Their Forms

There are several important forms of linear equations you need to know:

Standard Form: Ax + By = C (where A, B, C are integers and A ≥ 0)
Point-Slope Form: y - y₁ = m $x - x₁$ (where m is the slope and (x₁,y₁) is a point on the line)
Slope-Intercept Form: y = mx + b $where m is the slope and b is the y-intercept$

💡 When trying to determine if an equation is linear, look for variables with exponents other than 1 or variables multiplied together (like xy) - these make the equation non-linear!

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

Working with Linear Equations

Let's analyze the equation 3x - y = 8:

In standard form, we identify that A = 3, B = -1, and C = 8. Finding intercepts helps us graph the equation and understand its behavior.

To find the x-intercept, set y = 0 and solve for x: 3x - 0 = 8 3x = 8 x = 8/3 So the x-intercept is (8/3, 0)

For the y-intercept, set x = 0 and solve for y: 3(0) - y = 8 -y = 8 y = -8 So the y-intercept is (0, -8)

⚠️ Be careful with signs when rearranging equations! A common mistake is forgetting to change signs when moving terms from one side to another.

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

Converting Between Forms

When working with fractions in linear equations, you can multiply all terms by a common denominator to eliminate fractions. This makes the equation easier to work with.

For example, with ¼x + ⅘y = 4, we can multiply all terms by 20 (the LCM of 4 and 5): 5x + 16y = 80

Now we can easily identify the standard form values: A = 5, B = 16, C = 80.

Finding intercepts follows the same process as before:

For the x-intercept: set y = 0, which gives us 5x = 80, so x = 16, making the point (16, 0)
For the y-intercept: set x = 0, which gives us 16y = 80, so y = 5, making the point (0, 5)

🌟 Converting equations to standard form makes them easier to graph and analyze, especially when dealing with fractions!

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

Converting to Standard Form

Converting different equation types to standard form follows specific steps depending on the original form.

For equations like x - 3 = 9, first rearrange to isolate the variable: x - 3 = 9 x = 12 This is now in the form x = 12, which technically isn't standard form, but tells us the line is vertical at x = 12.

For equations with decimals like -0.15x - 0.72 = 19y, multiply by 100 to eliminate decimals: -15x - 72 = 19y Then rearrange to standard form: 15x + 19y = -72

🔑 When converting to standard form, remember that A should be positive. If you get a negative A value, multiply the entire equation by -1 to fix it!