My Car shakes

I have a 2000 Hyundai Elantra GLS. When i am going anywhere over 60 Miles an hour but then decellerate or break my car shakes until i get down to about 50. even when i am not touching the break just letting it coast back down it shakes. Anyone have any ideas as to why?

First note if vibration is in steering wheel, floor where your feet rest, or seat, or combination thereof, and write it down so later you’ll know. If your tires are ready for rotation, do that, except have all 4 balanced w/the stipulation that the mechanic check to see how far each tire is out of balance BEFORE he actually re-balances it and also note any tire or wheel out-of-roundness or side to side wobble. Just an

eyeball check at this point. If tires aren’t ready for rotation, follow above advice except don’t rotate! Duh. At this time mechanic should do a quick check for looseness, or binding, etc, in halfshafts and quick check of steering/suspension- loose ball joints or tie rods for ex. Then see if symptoms are gone, and take note of the following, but NOT during braking: 1.Vibration in steering wheel present? 2.Howbout floor?

3.Seat? Now, if these vibrations are gone, does it still vibrate during braking: Try various speeds and note as best you can if BRAKING vibration comes from steering wheel, floor, or seat, or combo thereof and write down. Maybe there won’t be any vibration now. Now w/info you wrote down mechanic can determine if you need brake rotor(s); or if rotor(s) can be machined; unless, of course the brake vibrations are now gone.