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King County has a population of 157,800157,800, and its population has been decreasing by 800800 people each year. Nearby Prince County has a population of 140,300140,300, and its population has been increasing by 1,7001,700 people each year. If these trends continue, how long will it take for the two counties to have the same population?

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Q. King County has a population of 157,800157,800, and its population has been decreasing by 800800 people each year. Nearby Prince County has a population of 140,300140,300, and its population has been increasing by 1,7001,700 people each year. If these trends continue, how long will it take for the two counties to have the same population?
  1. Population Data: King County population: 157,800157,800
    Decreasing by 800800 people/year
    Prince County population: 140,300140,300
    Increasing by 1,7001,700 people/year
  2. Population Equality: Let xx be the number of years it takes for the populations to be equal.
  3. Combine Terms: King County population after xx years: 157,800−800x157,800 - 800x Prince County population after xx years: 140,300+1,700x140,300 + 1,700x
  4. Solve for x: Set the populations equal to each other: 157,800−800x=140,300+1,700x157,800 - 800x = 140,300 + 1,700x
  5. Solve for x: Set the populations equal to each other: 157,800−800x=140,300+1,700x157,800 - 800x = 140,300 + 1,700x Combine like terms: 157,800−140,300=1,700x+800x157,800 - 140,300 = 1,700x + 800x
  6. Solve for x: Set the populations equal to each other: 157,800−800x=140,300+1,700x157,800 - 800x = 140,300 + 1,700x Combine like terms: 157,800−140,300=1,700x+800x157,800 - 140,300 = 1,700x + 800x 17,500=2,500x17,500 = 2,500x
  7. Solve for x: Set the populations equal to each other: 157,800−800x=140,300+1,700x157,800 - 800x = 140,300 + 1,700x Combine like terms: 157,800−140,300=1,700x+800x157,800 - 140,300 = 1,700x + 800x 17,500=2,500x17,500 = 2,500x Solve for x: x=17,5002,500x = \frac{17,500}{2,500}
  8. Solve for x: Set the populations equal to each other: 157,800−800x=140,300+1,700x157,800 - 800x = 140,300 + 1,700x Combine like terms: 157,800−140,300=1,700x+800x157,800 - 140,300 = 1,700x + 800x 17,500=2,500x17,500 = 2,500x Solve for x: x=17,5002,500x = \frac{17,500}{2,500} x=7x = 7

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