The multiple zeta values (MZVs) form a set of real numbers with, what we believe and module Zagier's Conjecture, a beautiful structure as an algebra over the rational numbers. In this paper we show partial solutions to five well known conjectures about the MZV obtained by using linear systems and shuffle products. We use computer algebra systems to study these conjectures and make our code available online. We describe how to compute the shuffle minus stuffle product to obtain relations of the MZVs, together with the xtaylor algorithm, to verify the conjectures mentioned above. We note that it is only necessary to solve a linear system, not in MZV but in its powers and products, to find all the relations. We also made all the code available online.