Q: cannot delete synced photos from iphone5

Hi there Shauna,

It sounds like you are unable to remove these synced photos from your phone, even if you uncheck the Photos checkbox in iTunes. I would quit the Photos, and Settings apps on your phone first:

Force an app to close in iOS

1. Press the Home button two times quickly. You'll see small previews of your recently used apps.
2. Swipe left to find the app you want to close.
3. Swipe up on the app's preview to close it.

 
When you have done that restart the device as well as your computer, and test the issue again:

iOS: Turning off and on (restarting) and resetting
 
If the issue persists, backup your device to iTunes, then restore it as new and verify that it works. 
 
How to erase your iOS device and then set it up as a new device or restore it from backups
 

If it does, then restore your backup to either verify it still works and the software just needed reinstalled, or isolate the issue to the backup file itself.  Thank you for using Apple Support Communities.