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?

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Solve a system of equations using any method: word problems

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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?

Define Variables: Let's define the variables: $\newline$ Let $x$ be the driving distance in kilometers. $\newline$ Let $y$ be the total fare in dollars. $\newline$ The first service charges $\$29$ up front plus $\$1$ per kilometer, so the total fare for the first service can be represented by the equation: $\newline$ $y = 1x + 29$ $\newline$ The second service charges $\$7$ up front plus $\$2$ per kilometer, so the total fare for the second service can be represented by the equation: $\newline$ $y = 2x + 7$ $\newline$ We need to find the value of $x$ at which $y$ is the same for both services.
Equations Representation: Now we have a system of two equations: $\newline$ $1$ . $y = 1x + 29$ $\newline$ $2$ . $y = 2x + 7$ $\newline$ Since both equations equal $y$ , we can set them equal to each other to find the value of $x$ : $\newline$ $1x + 29 = 2x + 7$
Solve Equations: Next, we solve for $x$ :
Subtract $1x$ from both sides to get:
$29 = x + 7$
Now, subtract $7$ from both sides to isolate $x$ :
$x = 29 - 7$
$x = 22$
So, the driving distance at which both services charge the same total fare is $22$ kilometers.
Substitute Value: Now that we have the value of $x$ , we can substitute it back into either of the original equations to find the total fare $y$ . Let's use the first equation: $\newline$ $y = 1x + 29$ $\newline$ $y = 1(22) + 29$ $\newline$ $y = 22 + 29$ $\newline$ $y = 51$ $\newline$ So, the total fare at which both services charge the same amount is $\$51$ .