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Proving an identity to have solutions over all the integers

  1. Aug 27, 2010 #1
    Hello,

    I was looking at some math problems and one kind caught my attention. The idea was to prove that lets 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?
     
  2. jcsd
  3. Aug 27, 2010 #2

    Office_Shredder

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    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?
     
  4. Aug 27, 2010 #3
    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?
     
  5. Aug 27, 2010 #4

    Office_Shredder

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    That looks good
     
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