Number of solutions of a linear 3x4 equation system

Question

If I have an inhomogeneous linear equation system with four variables and three equations, how do I see that there are either no solutions or infinitely many? I thought like this: The solution set of the corresponding homogeneous system is a linear subspace of $\mathbb{R}^4$ with dimension $0$ or $1$, that is the zero subset and a line through the origin. Then the solution set of the inhomogeneous system will be a translation of this solution set, hence the empty set or a line not through origin.