On Rado's boundedness conjecture

We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a1, a2, and a3 are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a1x1 + a2x2 + a3x3=0, then for every finite coloring of the positive integers there is a monochromatic solution to a1x1 + a2x2 + a3x3=0.