Question

according to newton's third law, when a horse pulls on a cart, the cart pulls back on the horse with an equal force on the horse. if in fact the cart pulls back on the horse as hard as the horse pulls forward on the cart, how is it possible for the horse to move the cart?

Answer 1

Based on Newton's principle, whenever objects A and B interact with each other, they exert forces upon each other.

When a horse pulls on a cart, the horse exerts a force only to the cart. But that force applies only to the cart, not to the horse.
 
The cart in turn exerts a force on the horse. But that force applies only to the horse, not the cart also.

There are two forces resulting from this interaction - a force on the horse and a force on the cart. The net force on the cart remains as it was --- a positive force in the direction of the horse's movement. Therefore, the cart begins to accelerate and move.