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Changing the equilibrium means that you are taking a system that is in equilibrium and removing a product/reactant so that a reaction could continue to happen. On the other hand, changing the equilibrium constant means that one would need to alter the pressure or temperature of the system as a whole to achieve the wanted reaction. It is possible to shift the equilibrium both by changing the equilibrium constant or not doing so as the equilibrium constant is dependent on pressure or temperature.
To my understanding, the equilibrium constant does not change— the ratio of products and reactants should always be the same at equilibrium. The only exception is that K will have a different value depending on the conditions (ex. temperature) that the reaction occurs under.
Changing temperature in a system in equilibrium will change the equilibrium constant. Changing the concentration or volume or pressure in a system in equilibrium would not change the equilibrium constant because the ratio between products and reactants would remain the same.
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