A simple solution approach is as a two step problem:
- Determine the magnitude of Force F that keeps the original 6m beam
- With Force F known, determine the shifted position of Support A that will maintain static equilibrium of the shortened 3.5 beam.
Before we jump into writing equations, another good step is to draw a free-body diagram of the beam and the applied forces. This will make it far easier to develop meaningful equations. For example, your proposed equation omits the self-weight of the 100 kg beam and drawing a FBD will help prevent errors like that.
By using variables to represent the beam length and support position, we can use the same FBD to write equations for both beam configurations.
First, in the original system, we know $x_1$, $x_2$, and $L$. We then solve for Force F by summing moments about Point A.
Second, in the shortened length system, we know Force F and Shortened Length L, and solve for the modified $x_1$ by once again summing moments about Point A.
Because this looks like a homework assignment, I'm not going to write out the equations but conceptually this is the approach I'd use.
Supplemental Thought: A nice zeroeth step is to consider what your intuition tells you. Imagine balancing a bar on your fingertip with your finger positioned like Support A (or, heck, go find a ruler and do an experiment!). Think about how you need to apply Force F to keep the bar stable. Now, imagine suddenly cutting off a chunk of the right end of the bar. What will happen to the system if Force F is kept constant? How might you shift your finger to keep that from happening? Taking a minute to do a thought experiment both develops intuition and serves as a gut-check on your final calculated answer.