The book shows the following :

Deriving a general node voltage method, we write a pair of KCL equations in terms of
unknown node voltages V​

​

​

​

1 and V2 this one time. We do this to illustrate a pattern for writing
equations by inspection.​

G​

​

​

​

AE1 + G3(E1 - E2) = I1 (1)​

GBE2 - G3(E1 - E2) = I2 (2)

Can you please explain how you derived (E1-E2) in each of the equations?

Thank you

 

The E1 and E2 is nothing like node voltages at V1 and V2.
So E1-E2 is voltage drop across G3 (R3)
http://www.allaboutcircuits.com/vol_1/chpt_10/4.html