I finally found the fix to the error in SolidWorks 2010 (also reported to work for SW 2011, 2015 and 2016) that prevented me from running the program. The computer uses Windows 7 32-bit (works also for Windows 8 and 10 as pointed out in the comments), and every time I started up the program it would something like:
The application failed to start because its side-by-side configuration is incorrect. Please see the application event log or use the command-line sxstrace.exe tool.
I used my Google-fu and found that the cause of this error is due my computer not having the Microsoft Visual C++ 2005 Redistributable. A copy of the setup file for this is in the SolidWorks 2010 disc. It is in the folder (DVD Drive):PreReqsVCRedist. After installing that, the error disappeared and SolidWorks starts properly.