Electronic – Mesh analysis with common voltage source

This is how i think i would solve the problem:

I don't see it necessary to add a 1V excitation source since there already a 9v in the circuit. The trick when analyzing circuit with dependent sources is to avoid the Short circuiting you would normally do when having a circuit with independent sources alone:

_Things to note that may help

Focus on node A:

define a current from the 9v source direction to node A and also define a current from 6ohm to A

_not Vx =Va

9 - Vx = I1

Vx/6 = I2

and i reckon Vab = 2Vx..

Hope this helps you!

First hint refer to diagram below, can you find the voltage drop across R3. If the values of all R and V is given.

Edit : Second hint refer to your original diagram, (I3 - I1)(30) + I3(20) + (I3 - I2)(40) = 0. Refer to diagram below for the direction of the current assumed. Try browsing through this link for better understandings.

Example:

Refer to the picture below if you should get the correct equation by following the steps in the image Image Source

I feel that you are asking a fundamental question in electrical engineering, that's only reason for me not to give you a direct answer or solution instead I'll try my best to guide you to get the solution.