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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineVijay and his good buddy Doug are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Vijay has already finished 33 oil changes today, and can complete more at a rate of 33 oil changes per hour. Doug just came on shift, and can finish 44 oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How long will that take? How many oil changes will Vijay and Doug each have done?\newlineIn _\_ hours, both men will have completed _\_ oil changes.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineVijay and his good buddy Doug are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Vijay has already finished 33 oil changes today, and can complete more at a rate of 33 oil changes per hour. Doug just came on shift, and can finish 44 oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How long will that take? How many oil changes will Vijay and Doug each have done?\newlineIn _\_ hours, both men will have completed _\_ oil changes.
  1. Define Variables: Define the variables for the system of equations.\newlineLet xx represent the number of hours since Doug started his shift, and yy represent the total number of oil changes completed by each person.
  2. Vijay's Equation: Write the equation for Vijay.\newlineVijay has already completed 33 oil changes and can do 33 more per hour. So, his equation based on the rate of completing oil changes is:\newliney=3x+3y = 3x + 3
  3. Doug's Equation: Write the equation for Doug.\newlineDoug is just starting and can complete 44 oil changes per hour. So, his equation is:\newliney=4xy = 4x
  4. Set Up System: Set up the system of equations.\newlineThe system of equations representing the situation is:\newliney=3x+3y = 3x + 3\newliney=4xy = 4x
  5. Solve by Substitution: Solve the system using substitution.\newlineSince both equations are equal to yy, set them equal to each other to find xx:\newline3x+3=4x3x + 3 = 4x
  6. Solve for x: Solve for x.\newlineSubtract 3x3x from both sides of the equation:\newline3x+3−3x=4x−3x3x + 3 - 3x = 4x - 3x\newline3=x3 = x
  7. Find yy: Find the value of yy by substituting xx into one of the original equations.\newlineUsing Doug's equation y=4xy = 4x and substituting x=3x = 3:\newliney=4(3)y = 4(3)\newliney=12y = 12
  8. Verify Solution: Verify the solution by substituting xx into Vijay's equation.\newlineUsing Vijay's equation y=3x+3y = 3x + 3 and substituting x=3x = 3:\newliney=3(3)+3y = 3(3) + 3\newliney=9+3y = 9 + 3\newliney=12y = 12\newlineSince the value of yy is the same for both equations, the solution is correct.

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