Write a system of equations to describe the situation below, solve using elimination, 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 $14$ tables and $22$ booths, which will seat a total of $262$ people. The second plan consists of $14$ tables and $21$ booths, which will seat a total of $252$ people. How many people can be seated at each type of table? $\newline$ Every table can seat $\_$ people, and every booth can seat $\_$ people.

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, 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

14

tables and

22

booths, which will seat a total of

262

people. The second plan consists of

14

tables and

21

booths, which will seat a total of

252

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 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 $B$ . We can then write two equations based on the given information: $\newline$ For the first plan: $14T + 22B = 262$ $\newline$ For the second plan: $14T + 21B = 252$ $\newline$ These two equations form our system of equations that we need to solve.
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 $T$ : $(14T + 22B) - (14T + 21B) = 262 - 252$
Subtract Equations: Performing the subtraction gives us: $\newline$ $14T - 14T + 22B - 21B = 262 - 252$ $\newline$ $0T + B = 10$ $\newline$ This simplifies to: $\newline$ $B = 10$ $\newline$ So, each booth can seat $10$ people.
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: $\newline$ $14T + 21B = 252$ $\newline$ $14T + 21(10) = 252$
Substitute B: Now we solve for $T$ : $14T + 210 = 252$ $14T = 252 - 210$ $14T = 42$
Solve for T: Divide both sides by $14$ to find T: $\newline$ $T = \frac{42}{14}$ $\newline$ $T = 3$ $\newline$ So, each table can seat $3$ people.