First what version of LR are you using?
In version 5 and below the default backup location is in a Backup folder with your original catalog. These backup have the extension .lrcat and can be opened directly in LR.
In version 6 and above the backup default location is the same but they are ZIP files and need to be extracted, expanded, before you can open them in LR.
The extension is .lrcat.zip.
I'm using version 5.
Do you know what the backup folder is called? I'm not finding it. The only thing I'm seeing is the folder under "Pictures" that has the lrcat and lr data files (one catalog, one previews).
Would they be filed as something else?
Do a search of your hard drive for .lrcat.
The default backup folder is called Backups, IIRC (I have changed the location on both my computers and renamed the folders for the 2 versions of LR I have installed to Lightroom 5 backups and Lightroom 6 backups), So I'm not completely sure what the original backup folder is called. But by default it is in the same folder as the LR catalog file. That default location for the catalog files is C:\ (for Windows) or Macintosh HD (for OS X) Users-YourUserName then either Document-Pictures-Lightroom (Windows) or Pictures-Lightroom (Mac).