Fourier–Motzkin elimination
Imagine you have a very big box full of toys of different shapes and sizes. Now, your teacher asks you to arrange all the toys in the box in a line according to the sizes, from small to big. But you can't see inside the box and you can only take one toy out at a time.
This is what Fourier-Motzkin elimination does for you. It helps you order variables in a system of inequalities (which is like a bunch of mathematical toys of different sizes) by looking at one variable at a time.
Let's say we have a few inequalities like:
x + y <= 3
2x + 3y <= 8
4x - y <= 1
We want to solve this system which means we want to find all the possible values of x and y that satisfy all the inequalities at the same time.
Now, we can use Fourier-Motzkin elimination to simplify this system by looking at one variable at a time. Let's say we start with x. We can isolate x in each inequality, so that we have:
x <= 3 - y
x <= (8 - 3y)/2
x >= (1 + y)/4
Now, we have three new inequalities in terms of x and y. And we can repeat this process for every variable in the system. Once we have gone through all the variables, we will have a much simpler system where each inequality only involves one of the variables.
This new system will tell us the same thing as the original system - which values of x and y can satisfy all the inequalities at the same time. But it will be easier to solve because we only have to look at one variable at a time.
In summary, Fourier-Motzkin elimination is a way of simplifying a system of inequalities by looking at one variable at a time and creating new inequalities that only involve that variable. This makes the system easier to solve and helps us find all the possible solutions.
This is what Fourier-Motzkin elimination does for you. It helps you order variables in a system of inequalities (which is like a bunch of mathematical toys of different sizes) by looking at one variable at a time.
Let's say we have a few inequalities like:
x + y <= 3
2x + 3y <= 8
4x - y <= 1
We want to solve this system which means we want to find all the possible values of x and y that satisfy all the inequalities at the same time.
Now, we can use Fourier-Motzkin elimination to simplify this system by looking at one variable at a time. Let's say we start with x. We can isolate x in each inequality, so that we have:
x <= 3 - y
x <= (8 - 3y)/2
x >= (1 + y)/4
Now, we have three new inequalities in terms of x and y. And we can repeat this process for every variable in the system. Once we have gone through all the variables, we will have a much simpler system where each inequality only involves one of the variables.
This new system will tell us the same thing as the original system - which values of x and y can satisfy all the inequalities at the same time. But it will be easier to solve because we only have to look at one variable at a time.
In summary, Fourier-Motzkin elimination is a way of simplifying a system of inequalities by looking at one variable at a time and creating new inequalities that only involve that variable. This makes the system easier to solve and helps us find all the possible solutions.