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 Sparrowtown Taxi Service. The first group requested $1$ sedan and $1$ minivan, which can seat a total of $10$ people. The second group asked for $3$ sedans and $1$ minivan, which can seat a total of $18$ 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 Sparrowtown Taxi Service. The first group requested

1

sedan and

1

minivan, which can seat a total of

10

people. The second group asked for

3

sedans and

1

minivan, which can seat a total of

18

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 $1$ minivan seating a total of $10$ people gives us the equation $s + m = 10$ .
Form Equations: The second group's request for $3$ sedans and $1$ minivan seating a total of $18$ people gives us the equation $3s + m = 18$ .
Eliminate Variable: We now have a system of two equations. To use elimination, we need to eliminate one of the variables, $s$ or $m$ . We choose to eliminate $m$ because it has the same coefficient in both equations.
Subtract Equations: We subtract the first equation from the second equation to eliminate $m$ . This gives us $3s + m - (s + m) = 18 - 10$ , which simplifies to $2s = 8$ .
Find Sedan Capacity: Dividing both sides of $2s = 8$ by $2$ , we find that $s = 4$ . This means a sedan can seat $4$ people.
Find Minivan Capacity: We substitute $s = 4$ into the first equation $s + m = 10$ and solve for $m$ . This gives us $4 + m = 10$ , which simplifies to $m = 6$ . This means a minivan can seat $6$ people.