Mesh Current method for solving a variable resistor?
From what I can see, your equations are correct. So all you need to do is set up the matrix using the first four equations, and get a Ia, Ib, Ic as a function of R0. 

Then simply substitute the value of Ic(R0) into the last equation to get the value of R0.

I tried it on Matlab, and got R0 = 40 or 40/49.