Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mr. Mitchell is contemplating which chauffeured car service to take to the airport. The first costs $\$29$ up front and $\$1$ per kilometer. The second costs $\$7$ plus $\$2$ per kilometer. For a certain driving distance, the two companies charge the same total fare. What is the distance? What is the total fare? $\newline$ For a driving distance of $\text{km}$ , the total fare is $\$\square$ .

Full solution

Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Mr. Mitchell is contemplating which chauffeured car service to take to the airport. The first costs

\$29

up front and

\$1

per kilometer. The second costs

\$7

plus

\$2

per kilometer. For a certain driving distance, the two companies charge the same total fare. What is the distance? What is the total fare?

\newline

For a driving distance of

\text{km}

, the total fare is

\$\square

Denote driving distance: Let's denote the driving distance in kilometers as $d$ . The total fare for the first service is $29 + 1d$ , and for the second service, it is $7 + 2d$ . We need to find the value of $d$ for which both services charge the same amount. So, we can set up the following system of equations: $\newline$ $1$ . $29 + d = 7 + 2d$ - This equation represents the point at which both services cost the same.
Set up system of equations: Now, we solve the equation for $d$ . We can subtract $d$ from both sides to get: $\newline$ $29 = 7 + d$ $\newline$ Then, we subtract $7$ from both sides to isolate $d$ : $\newline$ $22 = d$ $\newline$ So, the driving distance at which both services charge the same amount is $22$ kilometers.
Solve for driving distance: To find the total fare, we can substitute $d = 22$ into either of the original equations. Let's use the first service's equation: $\newline$ Total fare = $29 + 1 \times 22$ $\newline$ Total fare = $29 + 22$ $\newline$ Total fare = $51$ $\newline$ So, the total fare for both services at a driving distance of $22$ kilometers is $$51$.