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 Booneville Taxi Service. The first group requested $2$ sedans and $2$ minivans, which can seat a total of $18$ people. The second group asked for $3$ sedans and $2$ minivans, which can seat a total of $22$ 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 Booneville Taxi Service. The first group requested

2

sedans and

2

minivans, which can seat a total of

18

people. The second group asked for

3

sedans and

2

minivans, which can seat a total of

22

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

\newline

A sedan can seat _____ people, and a minivan can seat _____ people.

Define variables: Define the variables for the number of people each type of taxi can seat. $\newline$ Let's let $S$ represent the number of people a sedan can seat, and $M$ represent the number of people a minivan can seat.
Write equations: Write the system of equations based on the given information. $\newline$ For the first group: $2S + 2M = 18$ (Equation $1$ ) $\newline$ For the second group: $3S + 2M = 22$ (Equation $2$ )
Use elimination method: Use the elimination method to solve the system of equations. $\newline$ To eliminate one of the variables, we can multiply Equation $1$ by $-1.5$ and then add it to Equation $2$ . $\newline$ $-1.5(2S + 2M) = -1.5(18)$ $\newline$ $-3S - 3M = -27$ (Equation $3$ ) $\newline$ Now add Equation $3$ to Equation $2$ : $\newline$ $(3S + 2M) + (-3S - 3M) = 22 + (-27)$ $\newline$ $-1M = -5$
Solve for M: Solve for M, the number of people a minivan can seat. $\newline$ $M = 5$ $\newline$ Now that we know $M$ , we can substitute it back into one of the original equations to find $S$ .
Substitute $M$ for $S$ : Substitute $M$ into Equation $1$ to find $S$ . $\newline$ $2S + 2(5) = 18$ $\newline$ $2S + 10 = 18$ $\newline$ $2S = 18 - 10$ $\newline$ $2S = 8$ $\newline$ $S = 4$
Check solution: Check the solution by substituting $S$ and $M$ into Equation $2$ . $\newline$ $3(4) + 2(5) = 22$ $\newline$ $12 + 10 = 22$ $\newline$ $22 = 22$ $\newline$ The solution checks out.