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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineThe owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of 1515 tables and 2020 booths, which will seat a total of 205205 people. The second plan consists of 2525 tables and 2525 booths, which will seat a total of 275275 people. How many people can be seated at each type of table?\newlineEvery table can seat _\_ people, and every booth can seat _\_ people.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineThe owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of 1515 tables and 2020 booths, which will seat a total of 205205 people. The second plan consists of 2525 tables and 2525 booths, which will seat a total of 275275 people. How many people can be seated at each type of table?\newlineEvery table can seat _\_ people, and every booth can seat _\_ people.
  1. Define Variables: Let's denote the number of people each table can seat as tt and each booth can seat as bb. The first plan has 1515 tables and 2020 booths seating 205205 people, leading to the equation 15t+20b=20515t + 20b = 205.
  2. First Plan Equation: The second plan has 2525 tables and 2525 booths, seating 275275 people. This gives us the equation 25t+25b=27525t + 25b = 275.
  3. Second Plan Equation: Simplify the second equation by dividing all terms by 2525, resulting in t+b=11t + b = 11.
  4. Simplify Second Equation: To eliminate one variable, we'll eliminate bb. Multiply the simplified second equation by 2020 (the coefficient of bb in the first equation) to align the coefficients of bb. This results in 20t+20b=22020t + 20b = 220.
  5. Eliminate Variable: Subtract the first equation from this new equation: 20t+20b20t + 20b - 15t+20b15t + 20b = 220−205220 - 205, simplifying to 5t=155t = 15.
  6. Solve for tt: Solve for tt by dividing both sides by 55: t=3t = 3.
  7. Substitute to Find b: Substitute t=3t = 3 back into the simplified second equation t+b=11t + b = 11 to find bb: 3+b=113 + b = 11, so b=8b = 8.

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