Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ One Friday night, two large groups of people called Princeton Taxi Service. The first group requested $1$ sedan and $2$ minivans, which can seat a total of $15$ people. The second group asked for $2$ sedans and $3$ minivans, which can seat a total of $24$ people. How many passengers can each type of taxi seat? $\newline$ A sedan can seat $\_$ people, and a minivan 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

One Friday night, two large groups of people called Princeton Taxi Service. The first group requested

1

sedan and

2

minivans, which can seat a total of

15

people. The second group asked for

2

sedans and

3

minivans, which can seat a total of

24

people. How many passengers can each type of taxi seat?

\newline

A sedan can seat

\_

people, and a minivan can seat

\_

people.

Define Variables: Let's denote the number of people a sedan can seat as $s$ and the number of people a minivan can seat as $m$ . The first group's request for $1$ sedan and $2$ minivans seating a total of $15$ people gives us the equation $s + 2m = 15$ .
Form Equations: The second group's request for $2$ sedans and $3$ minivans seating a total of $24$ people gives us the equation $2s + 3m = 24$ .
Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, $s$ or $m$ . We choose to eliminate $s$ because its coefficients are $1$ and $2$ , which are easier to work with for elimination.
Multiply and Subtract: To eliminate $s$ , we multiply the first equation by $2$ , the coefficient of $s$ in the second equation. This gives us the new equation $2s + 4m = 30$ .
Solve for Minivan Capacity: We now subtract the first equation from the new second equation to eliminate $s$ . This gives us $2s + 4m - (2s + 3m) = 30 - 24$ , which simplifies to $m = 6$ .
Calculate Sedan Capacity: We substitute $m = 6$ into the first equation and solve for $s$ . This gives us $s + 2(6) = 15$ , which simplifies to $s + 12 = 15$ and then $s = 3$ .
Calculate Sedan Capacity: We substitute $m = 6$ into the first equation and solve for $s$ . This gives us $s + 2(6) = 15$ , which simplifies to $s + 12 = 15$ and then $s = 3$ . Therefore, the cost of each sedan is to seat $3$ people and each minivan is to seat $6$ people.