I checked the benchmarks for your setup. For Sims 4 gameplay it is not that bad. Your PC has enough power to run the game at the very least on Low graphic settings.
I also checked to see what type of games people are running on the AMD A6-5400K APU + AMD Radeon HD 7540D setup. I found someone on YouTube that has the same setup as you (actually lesser memory) running Battlefield 4. Battlefield 4 is extra, super tough on computers. It needs a LOT faster computer to run when compared to the Sims 4 requirements. Here is the video I found: http://youtu.be/Gwgmj7NhgD8
So if that guy can run Battlefield 4 with your setup then your computer is definitely powerful to run it at least on Low graphic settings.
Now onto the hard part… figuring out what the problem is, why you get the black screen that says The Sims 4 is “Not Supported”.
I might have figured it out. You will get that black screen and error message in a lot of games if you do not have the proper graphics drivers installed or if you do not have the latest graphics drivers installed while trying to play a modern game like The Sims 4.
Go here: http://support.amd.com/en-us/download
* Install the latest graphics drivers for your computer
* After the graphics drivers are fully installed you MUST restart your computer for it to work 100%
* After the computer is restarted try to play the Sims 4 demo