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Yeah. Even if you add a lot of reactant or product, give the reaction enough time, and it will eventually settle back to equilibrium with the same ratios of product to reactant. There will be more of both product and reactant in this scenario, but the ratio of the 2 will remain the same.
Changing the temperature would change the K. This is because if a reaction is exothermic, then increasing temp will cause the reaction to shift towards the reactants, so K would be smaller. The opposite would be true for endothermic reactions, where an increase in temp would result in an increase in K.
The only time K is changed is when the temperature is changed (endo/exothermic reactions). For changes in concentration or pressure, the reaction will adjust so it can eventually go back to equilibrium and therefore remain at the same K value.
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