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