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567

Jul 1, 2025

3 pages

Solve Linear Systems: Graphing, Substitution, and Elimination Fun!

A comprehensive guide to solving linear systems using graphing, substitution,... Show more

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

Linear Systems: Elimination

The elimination method involves adding or subtracting equations to eliminate one variable, allowing you to solve for the remaining variable.

Steps for the elimination method:

  1. Add or subtract equations to eliminate one variable
  2. Solve for the remaining variable
  3. Substitute the value into one of the original equations

Example: Solving 2x + 3y = 11 and -2x + 5y = 13

  1. Add the equations to eliminate x: 8y = 24
  2. Solve for y: y = 3
  3. Substitute y = 3 into 2x + 3y = 11 to find x = 1 The solution is 1,31, 3

Highlight: You can multiply one or both equations by a constant to facilitate elimination.

Vocabulary: Consistent independent systems have exactly one solution, which can be found using the elimination method.

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

Linear Systems: Substitution

The substitution method involves isolating a variable in one equation and substituting it into the other equation.

Steps for the substitution method:

  1. Isolate a variable in one equation
  2. Substitute the expression into the other equation and solve
  3. Use the result to find the value of the other variable

Example: Solving y = 3x + 2 and x + 2y = 11

  1. Substitute y = 3x + 2 into x + 2y = 11
  2. Solve for x: x + 23x+23x + 2 = 11, resulting in x = 1
  3. Find y by substituting x = 1 into y = 3x + 2, giving y = 5 The solution is 1,51, 5

Highlight: The substitution method is particularly useful when one equation is already solved for a variable.

Vocabulary: A consistent independent system will have a unique solution that can be found using the substitution method.

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

Linear Systems: Graphing

Linear systems consist of two or more linear equations with the same variables. A solution to a linear system is an ordered pair that satisfies all equations in the system. When graphing, the intersection point of the lines represents the solution.

To solve a linear system using the graph and check method:

  1. Graph both equations on the same coordinate plane
  2. Estimate the coordinates of the intersection point
  3. Check the coordinates algebraically

Definition: A linear system with exactly one solution is called a consistent independent system.

Example: Solving the system -x + y = -7 and x + 4y = -8

  1. Graph both equations
  2. Identify the intersection point 4,34, -3
  3. Check: -4 + 3-3 = -7 and 4 + 43-3 = -8 The solution is 4,34, -3

Highlight: Always verify the solution by substituting it back into both original equations.



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?

You have the choice, we offer a 100% secure payment via PayPal or an Amazon gift card. There is no limit to how much you can earn.

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

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

 

Algebra 1

567

Jul 1, 2025

3 pages

Solve Linear Systems: Graphing, Substitution, and Elimination Fun!

A comprehensive guide to solving linear systems using graphing, substitution, and eliminationmethods. This resource covers the fundamental concepts of linear systems, including consistent independent systems, and provides step-by-step instructions for each solving technique. Students will learn how to graph... Show more

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

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 Systems: Elimination

The elimination method involves adding or subtracting equations to eliminate one variable, allowing you to solve for the remaining variable.

Steps for the elimination method:

  1. Add or subtract equations to eliminate one variable
  2. Solve for the remaining variable
  3. Substitute the value into one of the original equations

Example: Solving 2x + 3y = 11 and -2x + 5y = 13

  1. Add the equations to eliminate x: 8y = 24
  2. Solve for y: y = 3
  3. Substitute y = 3 into 2x + 3y = 11 to find x = 1 The solution is 1,31, 3

Highlight: You can multiply one or both equations by a constant to facilitate elimination.

Vocabulary: Consistent independent systems have exactly one solution, which can be found using the elimination method.

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

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 Systems: Substitution

The substitution method involves isolating a variable in one equation and substituting it into the other equation.

Steps for the substitution method:

  1. Isolate a variable in one equation
  2. Substitute the expression into the other equation and solve
  3. Use the result to find the value of the other variable

Example: Solving y = 3x + 2 and x + 2y = 11

  1. Substitute y = 3x + 2 into x + 2y = 11
  2. Solve for x: x + 23x+23x + 2 = 11, resulting in x = 1
  3. Find y by substituting x = 1 into y = 3x + 2, giving y = 5 The solution is 1,51, 5

Highlight: The substitution method is particularly useful when one equation is already solved for a variable.

Vocabulary: A consistent independent system will have a unique solution that can be found using the substitution method.

linear systems: graphing
A linear system, consists of two or more linear equations
in the same variables.
.
A solution of a linear system is

Linear Systems: Graphing

Linear systems consist of two or more linear equations with the same variables. A solution to a linear system is an ordered pair that satisfies all equations in the system. When graphing, the intersection point of the lines represents the solution.

To solve a linear system using the graph and check method:

  1. Graph both equations on the same coordinate plane
  2. Estimate the coordinates of the intersection point
  3. Check the coordinates algebraically

Definition: A linear system with exactly one solution is called a consistent independent system.

Example: Solving the system -x + y = -7 and x + 4y = -8

  1. Graph both equations
  2. Identify the intersection point 4,34, -3
  3. Check: -4 + 3-3 = -7 and 4 + 43-3 = -8 The solution is 4,34, -3

Highlight: Always verify the solution by substituting it back into both original equations.

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?

You have the choice, we offer a 100% secure payment via PayPal or an Amazon gift card. There is no limit to how much you can earn.

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

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