Sorry, misread your post. Deleted my first reply, wrote a new one.

I think you meant T(t) = Te + (T[0]-Te)e-rt .

How can I tell? At t=0, e-rt = 1, so T(0)=T[0] (T[0] is the initial temperature). At t=infinity (a long time in the future), e-r*infinity = 0, so T(infinity) = Te (Te is the equilibrium temperature)

So, you are given

T(120) = Te + (T[0]-Te)e^-r*120

T(240) = Te + (T[0]-Te)e^-r*240

Subtract these

T(120) - T(240) = (T[0]-Te) ( e^-r*120 - e^-r*240 )

divide T[0]-Te =[T(120) - T(240)]/( e^-r*120 - e^-r*240 )

Define Delta T = T[0]-Te (this is a known quantity, now)

Go back to one of the two time points,

T(120) = Te + Delta T * e^-r*120

Everything is known except Te.

Now, you have

Te = T(120) - Delta T * e^-r*120

you already found

T[0]-Te = DeltaT

So, now you know T[0]