If both supply and demand increase at the same time, will prices rise as well? I need to explain it in relationship to the supply and demand curves.

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This will depend on whether the supply and the demand rise at the same rate or whether one or the other rises more quickly.

If supply rises faster than demand, the equilibrium price of the good will go down.  The supply curve will move to the right a given amount. ...

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This will depend on whether the supply and the demand rise at the same rate or whether one or the other rises more quickly.

If supply rises faster than demand, the equilibrium price of the good will go down.  The supply curve will move to the right a given amount.  The demand curve will also move to the right, but not as much.  This will lead to a lower equilibrium price than before.

If demand rises faster than supply, the opposite of the above will be true.  Both curves still move to the right, but the demand curve moves more and therefore the price rises.

So this really depends on the degree to which each of the two curves on your graph moves.

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