Solutions of a particular equation

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In summary, the problem at hand involves finding a solution to the equation z = x + y + b, where x, y, and z are variables, and b is a positive integer. The integers {1,2,...4b+5} are divided into two classes, and it needs to be shown that at least one of the classes contains a solution to the equation. The individual has attempted to use induction on b, but is struggling to use the induction hypothesis to prove the result. They believe a different approach is needed, but are unsure of how to proceed. They have also noticed that the problem is a special case of a more general research problem, which has already been solved. They are seeking advice or hints from others.
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A-ManESL
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Hello all... I have a problem which I have been grappling with for some time. Let b be a positive integer and consider the equation z = x + y + b where x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in two classes. I wish to show that at least one of the classes contains a solution to the equation.

I have tried using induction on b. The case b = 1 has been solved entirely by me. But I cannot understand how to use the induction hypothesis to prove the result. The more I think of it, the more I feel that a different approach to the problem is needed, but I can't figure out what. It is sort of a special case of a research problem, which has been solved in a more general way. I have little experience of doing research on my own, and so will be glad if anyone can offer me any advice or hints. Thanks.
 
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A class C of {1,2,...,4b+5} contains a solution is equivalent to that x,y,z are elements of C ?
 
  • #3
A-ManESL said:
Hello all... I have a problem which I have been grappling with for some time. Let b be a positive integer and consider the equation z = x + y + b where x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in two classes. I wish to show that at least one of the classes contains a solution to the equation.

I have tried using induction on b. The case b = 1 has been solved entirely by me. But I cannot understand how to use the induction hypothesis to prove the result. The more I think of it, the more I feel that a different approach to the problem is needed, but I can't figure out what. It is sort of a special case of a research problem, which has been solved in a more general way. I have little experience of doing research on my own, and so will be glad if anyone can offer me any advice or hints. Thanks.
If you give us your solution for the case b = 1, maybe someone could generalize it for the other cases. Oddly with one less or one more variable, i.e. z = w+x + y +1 or z = x+1, there is a simple counterexample.
 
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FAQ: Solutions of a particular equation

1. What is a solution of an equation?

A solution of an equation is a value that satisfies the equation when substituted for the variable.

2. How do you find the solutions of an equation?

To find the solutions of an equation, you can use various methods such as substitution, elimination, or graphing. The method used will depend on the type of equation and the given information.

3. Can an equation have more than one solution?

Yes, an equation can have more than one solution. In fact, some equations have an infinite number of solutions.

4. What does it mean if an equation has no solution?

If an equation has no solution, it means that there is no value that can be substituted for the variable to make the equation true. This could happen if the equation is contradictory or if the solutions involve complex numbers.

5. How do solutions of an equation relate to its graph?

The solutions of an equation are the points where the graph of the equation intersects the x-axis. This means that the x-coordinate of these points will give the solution to the equation.

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