  Starting from the results of Charles Fefferman and Janos Koll\`ar in
Continuous Solutions of Linear Equations [1], we adopt a new approach based on
Fefferman's techniques of Glaeser refinement to show a more general result than
the one proved by Koll\`ar by using techniques from algebraic geometry.
Considering a system of linear equations with semialgebraic (not only
polynomial as in [1]) coefficients on R^n, we get a necessary and sufficient
condition for the existence of a continuous and semialgebraic solution on R^n.
This is different from what Fefferman and Luli obtained in Semialgebraic
Sections Over the Plane [3] since they stated their result for solutions of
regularity C^m on the plane R^2. More in depth, we prove that a continuous and
semialgebraic solution on Rn exists if and only if there is a continuous
solution i.e., if the Glaeser-stable bundle associated to the system has no
empty fiber.
