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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineMr. Mitchell is contemplating which chauffeured car service to take to the airport. The first costs $29\$29 up front and $1\$1 per kilometer. The second costs $7\$7 plus $2\$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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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineMr. Mitchell is contemplating which chauffeured car service to take to the airport. The first costs $29\$29 up front and $1\$1 per kilometer. The second costs $7\$7 plus $2\$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?
  1. Define Variables: Let's define the variables:\newlineLet xx be the driving distance in kilometers.\newlineLet yy be the total fare in dollars.\newlineThe first service charges $29\$29 up front plus $1\$1 per kilometer, so the total fare for the first service can be represented by the equation:\newliney=1x+29y = 1x + 29\newlineThe second service charges $7\$7 up front plus $2\$2 per kilometer, so the total fare for the second service can be represented by the equation:\newliney=2x+7y = 2x + 7\newlineWe need to find the value of xx at which yy is the same for both services.
  2. Equations Representation: Now we have a system of two equations:\newline11. y=1x+29y = 1x + 29\newline22. y=2x+7y = 2x + 7\newlineSince both equations equal yy, we can set them equal to each other to find the value of xx:\newline1x+29=2x+71x + 29 = 2x + 7
  3. Solve Equations: Next, we solve for xx:
    Subtract 1x1x from both sides to get:
    29=x+729 = x + 7
    Now, subtract 77 from both sides to isolate xx:
    x=29−7x = 29 - 7
    x=22x = 22
    So, the driving distance at which both services charge the same total fare is 2222 kilometers.
  4. Substitute Value: Now that we have the value of xx, we can substitute it back into either of the original equations to find the total fare yy. Let's use the first equation:\newliney=1x+29y = 1x + 29\newliney=1(22)+29y = 1(22) + 29\newliney=22+29y = 22 + 29\newliney=51y = 51\newlineSo, the total fare at which both services charge the same amount is $51\$51.

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