|"makes a eeeeeeee noise " likely the fan for the PSU or CPU. If it is the CPU fan , it might cause the motherboard to shut down immediately and easy to fix.|
When the PSU goes out it might be the only problem or could take all with it. That includes the key board and monitor. Trying to replace one item at a time does not work to good as the bad component can cause the new to go bad. I would take it to a small comp repair shop. Two e machines of my friend both the PSU and motherboard were bad, CPU and memory were good.
There are power supply testers , just google.