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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMr. Russo is contemplating which chauffeured car service to take to the airport. The first costs $3\$3 up front and $4\$4 per mile. The second costs $13\$13 plus $2\$2 per mile. For a certain driving distance, the two companies charge the same total fare. What is the distance? What is the total fare?\newlineFor a driving distance of _____ miles, the total fare is $\$_____.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMr. Russo is contemplating which chauffeured car service to take to the airport. The first costs $3\$3 up front and $4\$4 per mile. The second costs $13\$13 plus $2\$2 per mile. For a certain driving distance, the two companies charge the same total fare. What is the distance? What is the total fare?\newlineFor a driving distance of _____ miles, the total fare is $\$_____.
  1. Define Variables: Let's define the variables.\newlineLet xx be the driving distance in miles.\newlineLet yy be the total fare in dollars.
  2. First Car Service Equation: Write the equation for the first car service.\newlineThe first car service charges $3\$3 up front and $4\$4 per mile.\newlineSo the equation for the first car service is: y=4x+3y = 4x + 3
  3. Second Car Service Equation: Write the equation for the second car service.\newlineThe second car service charges $13\$13 up front and $2\$2 per mile.\newlineSo the equation for the second car service is: y=2x+13y = 2x + 13
  4. Set Equations Equal: Set the two equations equal to each other to find the distance at which the total fares are the same. 4x+3=2x+134x + 3 = 2x + 13
  5. Solve for x: Solve for x by subtracting 2x2x from both sides of the equation.\newline4x+3−2x=2x+13−2x4x + 3 - 2x = 2x + 13 - 2x\newline2x+3=132x + 3 = 13
  6. Isolate x Term: Subtract 33 from both sides of the equation to isolate the term with xx.\newline2x+3−3=13−32x + 3 - 3 = 13 - 3\newline2x=102x = 10
  7. Divide to Solve xx: Divide both sides of the equation by 22 to solve for xx.2x2=102\frac{2x}{2} = \frac{10}{2}x=5x = 5
  8. Substitute xx into First Equation: 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. Using the first car service's equation: y=4x+3y = 4x + 3 y=4(5)+3y = 4(5) + 3 y=20+3y = 20 + 3 y=23y = 23
  9. Check Solution: Check the solution by substituting xx into the second car service's equation to ensure it gives the same total fare yy.\newlineUsing the second car service's equation: y=2x+13y = 2x + 13\newliney=2(5)+13y = 2(5) + 13\newliney=10+13y = 10 + 13\newliney=23y = 23

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