r is called a parameter, you have 3 unknowns and 2 equations which if you recall from high school algebra, means you do not have enough info to find a solution

when we say z=r and y=3-r and x=1+2r what this really means is you have enough info to determine y=3-z and x=1+2z, but because we lack a 3rd equation you can let z be anything

So particular solutions are of the form (1+2r,3-r,r) where you pick some number for r and then plug it in