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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 1414 tables and 2222 booths, which will seat a total of 262262 people. The second plan consists of 1414 tables and 2121 booths, which will seat a total of 252252 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 1414 tables and 2222 booths, which will seat a total of 262262 people. The second plan consists of 1414 tables and 2121 booths, which will seat a total of 252252 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 that can be seated at a table as TT and the number of people that can be seated at a booth as BB. We can then write two equations based on the given information:\newlineFor the first plan: 14T+22B=26214T + 22B = 262\newlineFor the second plan: 14T+21B=25214T + 21B = 252\newlineThese two equations form our system of equations that we need to solve.
  2. Use Elimination: To use elimination, we want to eliminate one of the variables by subtracting one equation from the other. We can subtract the second equation from the first to eliminate TT:(14T+22B)−(14T+21B)=262−252(14T + 22B) - (14T + 21B) = 262 - 252
  3. Subtract Equations: Performing the subtraction gives us:\newline14T−14T+22B−21B=262−25214T - 14T + 22B - 21B = 262 - 252\newline0T+B=100T + B = 10\newlineThis simplifies to:\newlineB=10B = 10\newlineSo, each booth can seat 1010 people.
  4. Solve for B: Now that we know the value of B, we can substitute it back into one of the original equations to solve for T. Let's use the second plan's equation:\newline14T+21B=25214T + 21B = 252\newline14T+21(10)=25214T + 21(10) = 252
  5. Substitute B: Now we solve for TT:14T+210=25214T + 210 = 25214T=252−21014T = 252 - 21014T=4214T = 42
  6. Solve for T: Divide both sides by 1414 to find T:\newlineT=4214T = \frac{42}{14}\newlineT=3T = 3\newlineSo, each table can seat 33 people.

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