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Dec 7, 2025
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Understanding Linear Programming: Concepts and Examples
Isabel
@melonabread_
1 / 4
Linear Programming Basics
Linear programming starts with clearly labeling your variables and setting up your constraints as inequalities. These constraints create a feasible region where all possible solutions exist. The key is finding which point in this region gives you the best result.
To find the optimal solution, you need to identify the vertices (corner points) of your feasible region. This means solving pairs of equations where the constraint lines intersect. Once you have all vertices, plug each one into your optimization function to find where the maximum or minimum value occurs.
Let's see this in action with species conservation. If two bird species need different amounts of land and food, linear programming can determine the maximum number of each species the habitat can support. For example, if Species A needs 120m² of land and 39.6kg of food each, while Species B needs 90m² and 69.6kg, we can find the exact combination that maximizes conservation efforts.
Quick Tip: Always check all vertices of your feasible region when searching for the optimal solution - the best answer is always at one of the corners!
Applied Linear Programming Problems
When a company makes multiple products with shared resources, linear programming becomes essential. For instance, if a shoe manufacturer makes outdoor cleats requiring 2 hours in step 1 and 1 hour in step 2 (profit $20), and indoor cleats requiring 1 hour in step 1 and 3 hours in step 2 (profit $15), we can find the perfect production mix.
Setting up constraints based on available hours and calculating the profit function , we can determine the most profitable production plan. By testing each vertex of our feasible region, we find the point that maximizes profit. In our example, the optimal solution was 12 outdoor cleats and 16 indoor cleats, generating $480 profit.
Businesses use this same approach for inventory decisions. A toy seller who pays 2) and 3) can determine the optimal purchase quantities given constraints like monthly sales limits and investment capital. By testing all vertices in the feasible region, we found that selling 1333 of toy A and 666 of toy B maximizes profit at $4664.
Real-world Connection: Businesses use linear programming every day to maximize profits while managing limited resources like time, money, and production capacity!
Maximizing Storage with Linear Programming
Ever needed to make the most of limited space? Linear programming can help with everyday decisions like buying file cabinets. Imagine you need to buy vertical file cabinets ($10 each, 6 sq. ft. space, 8 cubic ft. storage) and lateral cabinets ($20 each, 8 sq. ft. space, 12 cubic ft. storage).
With constraints like a $140 budget and 72 sq. ft. of available floor space, you can set up inequalities to find the best combination. The space constraint gives us l ≤ -¾v + 9, while the budget constraint yields l ≤ -½v + 7, creating our feasible region.
To find the optimal solution, we identify the vertices of our feasible region: (0,0), (0,7), (12,0), and (6,3). Our goal is to maximize total storage volume represented by the function V = 8v + 12l.
When we evaluate each vertex in our optimization function, we're looking for the highest value. This systematic approach helps us make the most efficient use of our resources instead of guessing.
Study Strategy: Draw out your feasible region on graph paper to visualize the problem better. This makes it much easier to identify the vertices you'll need to test!
Finding the Optimal Solution
After identifying all vertices of our feasible region, we need to calculate the value of our optimization function at each point. For our file cabinet example, we evaluate V = 8v + 12l at each vertex:
At (0,0): 8(0) + 12(0) = 0 cubic feet At (0,7): 8(0) + 12(7) = 84 cubic feet At (12,0): 8(12) + 12(0) = 96 cubic feet At (6,3): 8(6) + 12(3) = 100 cubic feet
The highest value occurs at the point (6,3), giving us 100 cubic feet of storage. This means buying 6 vertical file cabinets and 3 lateral cabinets will maximize our storage capacity while staying within our budget and space constraints.
This methodical approach to decision-making is powerful because it gives us certainty that we've found the best possible solution. Rather than relying on intuition, we can mathematically prove we've optimized our resources.
Remember: The optimal solution in linear programming always occurs at a vertex of the feasible region, never in the middle of a region or on a line between vertices!
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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
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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
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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
Understanding Linear Programming: Concepts and Examples
Isabel
@melonabread_
Linear programming is a powerful technique for finding the best solution when there are limited resources or constraints. It helps you maximize or minimize a goal (like profit or cost) while staying within set boundaries. This is a skill you'll... Show more
Linear Programming Basics
Linear programming starts with clearly labeling your variables and setting up your constraints as inequalities. These constraints create a feasible region where all possible solutions exist. The key is finding which point in this region gives you the best result.
To find the optimal solution, you need to identify the vertices (corner points) of your feasible region. This means solving pairs of equations where the constraint lines intersect. Once you have all vertices, plug each one into your optimization function to find where the maximum or minimum value occurs.
Let's see this in action with species conservation. If two bird species need different amounts of land and food, linear programming can determine the maximum number of each species the habitat can support. For example, if Species A needs 120m² of land and 39.6kg of food each, while Species B needs 90m² and 69.6kg, we can find the exact combination that maximizes conservation efforts.
Quick Tip: Always check all vertices of your feasible region when searching for the optimal solution - the best answer is always at one of the corners!
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Improve your grades
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Applied Linear Programming Problems
When a company makes multiple products with shared resources, linear programming becomes essential. For instance, if a shoe manufacturer makes outdoor cleats requiring 2 hours in step 1 and 1 hour in step 2 (profit $20), and indoor cleats requiring 1 hour in step 1 and 3 hours in step 2 (profit $15), we can find the perfect production mix.
Setting up constraints based on available hours and calculating the profit function , we can determine the most profitable production plan. By testing each vertex of our feasible region, we find the point that maximizes profit. In our example, the optimal solution was 12 outdoor cleats and 16 indoor cleats, generating $480 profit.
Businesses use this same approach for inventory decisions. A toy seller who pays 2) and 3) can determine the optimal purchase quantities given constraints like monthly sales limits and investment capital. By testing all vertices in the feasible region, we found that selling 1333 of toy A and 666 of toy B maximizes profit at $4664.
Real-world Connection: Businesses use linear programming every day to maximize profits while managing limited resources like time, money, and production capacity!
Sign up to see the contentIt's free!
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Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Maximizing Storage with Linear Programming
Ever needed to make the most of limited space? Linear programming can help with everyday decisions like buying file cabinets. Imagine you need to buy vertical file cabinets ($10 each, 6 sq. ft. space, 8 cubic ft. storage) and lateral cabinets ($20 each, 8 sq. ft. space, 12 cubic ft. storage).
With constraints like a $140 budget and 72 sq. ft. of available floor space, you can set up inequalities to find the best combination. The space constraint gives us l ≤ -¾v + 9, while the budget constraint yields l ≤ -½v + 7, creating our feasible region.
To find the optimal solution, we identify the vertices of our feasible region: (0,0), (0,7), (12,0), and (6,3). Our goal is to maximize total storage volume represented by the function V = 8v + 12l.
When we evaluate each vertex in our optimization function, we're looking for the highest value. This systematic approach helps us make the most efficient use of our resources instead of guessing.
Study Strategy: Draw out your feasible region on graph paper to visualize the problem better. This makes it much easier to identify the vertices you'll need to test!
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
Finding the Optimal Solution
After identifying all vertices of our feasible region, we need to calculate the value of our optimization function at each point. For our file cabinet example, we evaluate V = 8v + 12l at each vertex:
At (0,0): 8(0) + 12(0) = 0 cubic feet At (0,7): 8(0) + 12(7) = 84 cubic feet At (12,0): 8(12) + 12(0) = 96 cubic feet At (6,3): 8(6) + 12(3) = 100 cubic feet
The highest value occurs at the point (6,3), giving us 100 cubic feet of storage. This means buying 6 vertical file cabinets and 3 lateral cabinets will maximize our storage capacity while staying within our budget and space constraints.
This methodical approach to decision-making is powerful because it gives us certainty that we've found the best possible solution. Rather than relying on intuition, we can mathematically prove we've optimized our resources.
Remember: The optimal solution in linear programming always occurs at a vertex of the feasible region, never in the middle of a region or on a line between vertices!
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