Why not try solving this like a pair of linear equations?

Multiply the 1st equation by 2 and subtract the 2nd to get:

10x = 3 mod p. This has a solution since (10,p)=1 so there exists a y such that 10 * y = 1(p). Then x = 3 * y (p).

Once you have x,

and again because (3,p)=1 you know that the inverse of 3 exists.