Proving an identity to have solutions over all the integers

[jimep]

Hello,

I was looking at some math problems and one kind caught my attention. The idea was to prove that let's say 3x+2y=5 has infinitely many solutions over the integers.

Can someone show me the procedure how a problem like this might be solved?

 

[Office_Shredder]

One solution is x=3, y=-2. Can you find a way to add some number a to 3 and subtract another number b from -2 so that when you plug 3+a and -2-b into the equation the a term and the b term cancel out?

 

[jimep]

Thank you man, I perfectly understood how to solve this.

3x+2y=5 => 3*(2a+1)+2*(1-3a)=5

And then just to practice I solved another one

4x+3y=10 => 4*(3n+1)+3*(2-4n)=10

Are they correct?

 

[Office_Shredder]

That looks good

 

[CellCoree]

I can provide a response to your question. Proving an identity to have solutions over all integers requires a technique called mathematical induction. This method involves proving the identity for a base case, typically the smallest integer value, and then showing that if the identity holds for a certain integer value, it also holds for the next integer value. This process is repeated until it can be shown that the identity holds for all integer values.

In your example, 3x+2y=5, you can start by choosing a base case, such as x=0 and y=5/2. This satisfies the equation and shows that the identity holds for at least one set of integer values. Next, you can assume that the identity holds for some integer values, say x=a and y=b, and then prove that it also holds for the next integer values, x=a+1 and y=b+1. This can be done by substituting these values into the equation and showing that it still holds true.

If you can successfully show that the identity holds for the base case and that it also holds for the next integer values, then you can conclude that it holds for all integer values. This means that there are infinitely many solutions for this identity over the integers.

I hope this explanation helps you understand the process of proving an identity to have solutions over all integers. Keep in mind that this is just one approach and there may be other methods to prove the same result. It is always important to carefully consider the assumptions and steps taken in any mathematical proof.