Howdy, and welcome to the forums!
The extension service did a study years ago that estimated it would take removing 75% of the population annually to make a dent in the population.
I've quit shooting coyotes as much as I did, as I've come to the conclusion they are likely more beneficial to me than they are detrimental. I generally have been leaving them be unless they are causing me problems. For things like quail, they are predators of quail predators, and since there are still some quail in my neck of the woods, I try to do what I can to help the quail. The quail population in Texas is suffering enough. They would cause problems for the goats, but the guard dogs prevent that. I've even watched them actively hunt coyotes that stray too close to the herd. I wish they would develop more of a craving for bacon and do something about all the hogs.