I have a shapefile that shows up in QGIS and ArcGIS correctly.
But when I inspect the attributes of the layer it does not have latitude or longitude.
I need to extract the coordinate system so I'm looking for the lat/longs using QGIS.
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That's what shapefiles do: keep the coordinates in a binary where you cannot see them. In qgis go to the field calculator, ad a field "x": double, length 12 with 8 precision, and then write "$x" in the calculator, the same for "y"; if you want the nodes of a polygon or a line, first extract them (there's an option in the toolbox), then do the field calculator procedure.
You can find the coordinate system in the 'general' tab of the layer properties dialog box (right click on the layer to bring up the properties option). You can also find coordinate system info in the .prj file of the shapefile (if you are indeed using a shapefile).