A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well known example is the Minimum Norm (MN) solution, which is the solution having the least Euclidean norm. In this note, we consider another useful solution, related with the MN one, which we call the Minimum Variance (MV) solution. We discuss some of its properties and derive a simple, closed form expression.