How has the blues influences Jazz music?

Jazz was a result of blues musicians modifying the time and key structures of their performances.
Elvis kicked blues into high gear by performing blues songs along with some rockabilly.
When the music industry saw how he affected girls they pushed to distribute more blues music.
Thank the British for reminding us how it was supposed to sound.

Blues, as a style, was always an undertone, meaning musicians knew about it, and took ideas from blues, (i.e. Chuck Berry would get a few ideas from blues musicians, then Elvis would get a few ideas from Chuck Berry) but most people didn’t… that is until the 1960s and 70s when the interest in blues by popular groups like Led Zeppelin and Eric Clapton brought then unknown blues musicians to the forefront. Now, everyone knows who BB King is.

So, when you’re imagining the blues that effected Jazz in it’s early development, it’s easy to get lost in rock-blues, like is performed by BB King. (it’s not really "rock blues" but it’s the blues that effected rock stars in that era) It’s closer to the blues as sung by guys like Leadbelly. (of course he was influenced by guys like WC Handy) Acoustic, (guitars/ banjos) slides, based on folk styles. In fact, 10, 16, and 20 bar formulas were common when blues was effecting Jazz in the late 19th Century, not just the standard "12 bar blues" we’ve come to know as the blues.

After the Civil War, many of the soldiers came back from the war to their towns with trumpets, trombones, drums and many other instruments that are now common to Jazz. They also had a love for the march music they played in the army camps… with the influx of these instruments, which were later sold in stores, then bought by young kids, most black or interracial (though not all… there have indeed by white kids involved in jazz from the start), and they tried to play the blues (then guitar and singing) on these marching wind instruments. What ended up coming out was jazz. (early Ragtime jazz) Even today, in New Orleans you can see Brass Bands jazz marching through the streets during Mardi Gras.

If Jazz is America’s "Classical Music," then blues would have to be the baroque music out of which it grew, and evolved into the complex, powerful form it is today. Blues itself grew out of Spirituals, sung in the fields during work hours to pass the time.