I need some closure on the following, Question: Prove that if a linear system of equations with only rational coefficients and constants has a solution then it has at least one all-rational solution. Must it have infinitely many? My Solution: If the RHS of the equations is a rational number, then the sum of the terms of the LHS must be rational, so the terms must be rational and hence the solution must be rational right? The only reason this argument wouldn't work is if the RHS is equal to 0. Must if have infinitely many rational solutions? No, but it can though (if the form of the solution includes some free variables, picking rational numbers for the free variables will produce a rational solution).