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This is possible because demand is a function of many many factors. The biggest ones are price and wealth (also known as income). If an agent's income goes up, their demand for any given good will also good up. If the price goes up, an agent's demand will go down. Thus you have the price and income elasticity of demand. In a market with two goods, if agents divide their income amongst goods (for instance apples and Oranges), you could easily derive a cross-price elasticity of demand that measures how much the price/demand of one good changes when the other good changes.

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Q: How is it possible for many elasticities to be associated with a single demand curve?
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