Hanging Shelf With 79lb Drywall Anchor
No those anchors will NOT be sufficient to hold up THAT shelf. Those shelf brackets are too short and there will be a LARGE pulling force on the upper screw. 

Assuming from the image that's a 10 inch shelf and the screws at 2 inches apart, hanging 79lbs on the edge of the shelf will get you a whopping 


  395 LBS of pull force on the top screw.


When you look at that picture and look at the metal end, does it remind you of any tool that is sitting on top of your toolbox? A claw-hammer perhaps?

You may get away with the type of anchor that mushrooms out behind the wall as shown below. Remembering to buy the size that matches your drywall thickness. But I do NOT recommend these either for that shelf.



With the shortness of that bracket, I'd still be concerned that was not enough to hold up much weight. If those fail, it is going to pull down a rather large piece of drywall too.

Personally I would be inclined to add a backer board out of a nice piece of wood, the wider the better, using one of the following two methods. 

Option 1: Outboard screws... Board cut to fit to next stud width. In your case 3 (Don't forget to add a 2x4 width.). Mount board to studs then centre shelf on the wood.



Option 2.: Inboard Screws. Cut board to about 1 inch wider than the shelf. Then centre the board on the studs as shown in the image below.



Which method is best? 

Use the method that gets the screws in the wood closest to the screws in the brackets. However, be aware that with the "inboard screws" method, the end of the board will tend to pull away from the wall if the screws are too far away from the brackets. 

If the boards spans more than two studs, I'd add another couple of screws  at each stud.

If you, or the Mrs, don't like the visible screws, use one of the many capping or plugging methods.