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 $15$ tables and $20$ booths, which will seat a total of $205$ people. The second plan consists of $25$ tables and $25$ booths, which will seat a total of $275$ 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

15

tables and

20

booths, which will seat a total of

205

people. The second plan consists of

25

tables and

25

booths, which will seat a total of

275

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 each table can seat as $t$ and each booth can seat as $b$ . The first plan has $15$ tables and $20$ booths seating $205$ people, leading to the equation $15t + 20b = 205$ .
First Plan Equation: The second plan has $25$ tables and $25$ booths, seating $275$ people. This gives us the equation $25t + 25b = 275$ .
Second Plan Equation: Simplify the second equation by dividing all terms by $25$ , resulting in $t + b = 11$ .
Simplify Second Equation: To eliminate one variable, we'll eliminate $b$ . Multiply the simplified second equation by $20$ (the coefficient of $b$ in the first equation) to align the coefficients of $b$ . This results in $20t + 20b = 220$ .
Eliminate Variable: Subtract the first equation from this new equation: $20t + 20b$ - $15t + 20b$ = $220 - 205$ , simplifying to $5t = 15$ .
Solve for $t$ : Solve for $t$ by dividing both sides by $5$ : $t = 3$ .
Substitute to Find b: Substitute $t = 3$ back into the simplified second equation $t + b = 11$ to find $b$ : $3 + b = 11$ , so $b = 8$ .