(1)  If TWO separate equations are presented in the post, then wording must be THIS :

         Solve equations  3x = 1(mod 7) and 5x = 1(mod 8)


     using plural  "equations".



(2)  If one entity is assumed, then wording must be THIS :

         Solve the system of two equations  3x = 1(mod 7) and 5x = 1(mod 8)


     using the term "system".

First equation,   3x = 1(mod 7),  is equivalent to  x = 5(mod 7).


Second equation,  5x = 1(mod 8),  is equivalent to  x = 5(mod 8).


Then the system is equivalent to one equation  x = 5(mod 56).


ANSWER.  The solution to the system is  x = 5(mod 56).