Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. The owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of 18 tables and 20 booths, which will seat a total of 290 people. The second plan consists of 20 tables and 25 booths, which will seat a total of 350 people. How many people can be seated at each type of table? Every table can seat _____ people, and every booth can seat _____ people.

Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.  The owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of 18 tables and 20 booths, which will seat a total of 290 people. The second plan consists of 20 tables and 25 booths, which will seat a total of 350 people. How many people can be seated at each type of table?  Every table can seat _____ people, and every booth can seat _____ people.

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Define Variables: Let's denote the number of people that can be seated at a table as t and the number of people that can be seated at a booth as b. The first plan consists of 18 tables and 20 booths seating a total of 290 people, which gives us the equation 18t + 20b = 290.First Plan: The second plan consists of 20 tables and 25 booths seating a total of 350 people, which gives us the equation 20t + 25b = 350.Second Plan: We now have a system of two equations. We need to eliminate one of the variables, t or b. We choose to eliminate t because its coefficients are close to each other, which might make the calculations simpler.Eliminate Variable: To eliminate t, we can multiply the first equation by 20 and the second equation by 18, so the coefficients of t in both equations are the same. This gives us the new equations 360t + 400b = 5800 (first equation multiplied by 20) and 360t + 450b = 6300 (second equation multiplied by 18).Subtract Equations: We now subtract the first new equation from the second new equation to eliminate t. This gives us 50b = 500.Find Booth Capacity: Solving for b, we divide both sides of the equation by 50, which gives us b = 10. This means that every booth can seat 10 people.Substitute in Equation: We substitute b = 10 into the first original equation 18t + 20b = 290. This gives us 18t + 20(10) = 290.Solve for Tables: Solving for t, we first calculate 20(10) which is 200, and then we subtract 200 from 290, which gives us 18t = 90.Final Solution: Finally, we divide both sides of the equation by 18 to find t, which gives us t = 5. This means that every table can seat 5 people.

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