Can't seem to figure out equations in Mesh Current Method
Because mesh 1 has a current source in it (not shared with another mesh) you can just write
$$I_1 = 10\ {\rm mA}$$ 
in place of your first equation.

Similarly, for mesh 6,
$$I_6 = 10\ {\rm mA}.$$ 

If either of these current sources had been on a branch shared by two meshes, you'd have had to introduce super-meshes.