Let v1 be the number of vertices of degree 4 and v2 be the number of vertices of degree 5.

We know that v1+v2=16 and from handshake lemma we get that 4v1+5v2=70.

So you only need to solve the system of 2 linear equations

v1+v2=16

4v1+5v2=70

Solution is v1=10, v2=6.