Why can you play a chord without a fifth?
This specific version of it's parent the dominant seventh chord has no perfect fifth. The main reason for this is to make it feel lighter.
Not playing a perfect fifth is very common in jazz, classical and world music. The perfect fifth itself does not add any importance to a chord in the sense that it does not make it minor, major or suspended. It does not create any tension but merely solidifies the tonic making the chord heavier. Not playing a perfect fifth does not alter the meaning of the chord in any way.
On some instruments, such as the guitar or ukulele, the fifth might be left out because it would otherwise be impossible to play a certain voicing. In other words: it is only left out because of a technical aspect of the instrument.
In nearly any musical style, you can leave the fifth out of many chords. Some of them are: any variant of a minor chord with extended notes (m7,m9,m11,m13) and major or dominant major chord with extended notes (9, 13, maj13, etc).
Some chord actually do need a fifth, for example while playing m7b5. Obviously enough, the fifth in that chord is altered downwards to a flat fifth. Giving significance and tension to the interval within that chord.
