Trading in financed car?

I have a 2016 Hyundai with 35k miles. I still owe $14,500 on the car. Is it possible to trade the car in for instance, a $9,000 used Honda, without owing money? My goal is to eliminate monthly car payments.

Not likely. You will end up with negative equity and be paying on a car that's not as good as what you had.

One way to eliminate monthly car payments is to pay off your loan. You can always ask a dealer yourself, especially if you have a specific car in mind, and they can go over your options with you, and if it's worth it.

No.
A dealer is not going to take this deal.

You haven't mentioned what your current car is worth. If it is worth more than $14500 you sell it and then you can purchase another car for cash.

Your issue is you are roughly $8000 upside down. That does not just go away because you don't want a car payment.

You lose thousands every time you trade.

Chances are you should keep what you have for a very long time.

A 2nd job will help you pay it off sooner but do NOT be in a hurry to change cars & lose thousands in value.

The $ 14500 will never go away. I will just be added to your next purchase. Doing this can really get you into deep sh_t in a very short time. You can wind up paying over $20,000 plus on your next car purchase.

Is it possible? Yes depending what they give you for your current vehicle. To pay off your current vehicle and not owe anything on the next car, they have to give you $23,500 for your existing vehicle.

Is it likely? No.

Consult Kelly Blue Book to see the value of your Hyundai. That would be wholesale blue book, not private party. Now deduct what you owe. Now deduct another $1000 because car companies don't do things for free. That is how much your Hyundai is worth. I'll bet it is a lot less than $9000.

There's no way it won't require you to pay some cash.

To eliminate your payment you would need to top up the amount you are underwater, about $8000 AND pay cash for the other car, All together it would be the exact same amount of cash needed just to pay off your car loan and you get to keep the newer car.

YOU owe $14,500. If you can sell the car for $23,500 then you can pay off the outstanding loan, have $9000 cash and can buy the Honda and not owe any money. I seriously doubt you can sell the car for that much because that is far more than you paid for it new. Best you can do is sell it, pay off the loan and go buy a cheap junker with whatever cash you have.