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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ 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 $11$ booths, which will seat a total of $182$ people. The second plan consists of $25$ tables and $11$ booths, which will seat a total of $210$ people. How many people can be seated at each type of table? $\newline$ Every table can seat _ people, and every booth can seat _ people.

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Solve a system of equations using any method: word problems

Full solution

Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

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

11

booths, which will seat a total of

182

people. The second plan consists of

25

tables and

11

booths, which will seat a total of

210

people. How many people can be seated at each type of table?

\newline

Every table can seat _____ people, and every booth can seat _____ people.

Define Equations: 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 $B$ . We can then write two equations based on the given information. $\newline$ First plan: $18T + 11B = 182$ $\newline$ Second plan: $25T + 11B = 210$
Create System: We have a system of linear equations: $\newline$ $18T + 11B = 182$ $\newline$ $25T + 11B = 210$ $\newline$ To solve the system, we can subtract the first equation from the second to eliminate $B$ . $\newline$ $(25T + 11B) - (18T + 11B) = 210 - 182$
Eliminate Variable: Perform the subtraction to find the value of $T$ . $\newline$ $25T - 18T + 11B - 11B = 210 - 182$ $\newline$ $7T = 28$ $\newline$ Now, divide both sides by $7$ to solve for $T$ . $\newline$ $T = \frac{28}{7}$ $\newline$ $T = 4$
Solve for T: Now that we have the value for $T$ , we can substitute it back into one of the original equations to solve for $B$ . Let's use the first plan's equation: $\newline$ $18T + 11B = 182$ $\newline$ $18(4) + 11B = 182$ $\newline$ $72 + 11B = 182$
Substitute and Solve: Subtract $72$ from both sides to solve for $B$ . $\newline$ $11B = 182 - 72$ $\newline$ $11B = 110$ $\newline$ Now, divide both sides by $11$ to find the value of $B$ . $\newline$ $B = \frac{110}{11}$ $\newline$ $B = 10$

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