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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineSanjay has punch cards for his favorite tea house and his favorite coffee shop. He currently has 66 punches on the tea punch card and 88 punches on the coffee punch card. Given his regular routine, he consistently earns 55 new punches per week on the tea punch card and 44 on the coffee punch card. Before too long, Sanjay will have the same number of punches on each card. How many punches will Sanjay have on each card? How long will that take?\newlineSanjay will have _\_ punches on each card in _\_ weeks.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineSanjay has punch cards for his favorite tea house and his favorite coffee shop. He currently has 66 punches on the tea punch card and 88 punches on the coffee punch card. Given his regular routine, he consistently earns 55 new punches per week on the tea punch card and 44 on the coffee punch card. Before too long, Sanjay will have the same number of punches on each card. How many punches will Sanjay have on each card? How long will that take?\newlineSanjay will have _\_ punches on each card in _\_ weeks.
  1. Define Variables: Let's define the variables for the number of punches on each card. Let TT represent the number of punches on the tea card and CC represent the number of punches on the coffee card. Initially, T=6T = 6 and C=8C = 8. Each week, Sanjay earns 55 punches on the tea card and 44 on the coffee card. We can write the equations as:\newlineT=6+5wT = 6 + 5w (Equation 11)\newlineC=8+4wC = 8 + 4w (Equation 22)\newlineWhere ww is the number of weeks.
  2. Set Equations Equal: We need to find when TT equals CC. So, set Equation 11 equal to Equation 22:\newline6+5w=8+4w6 + 5w = 8 + 4w\newlineNow, solve for ww:\newline5w−4w=8−65w - 4w = 8 - 6\newlinew=2w = 2
  3. Solve for ww: Substitute w=2w = 2 back into either Equation 11 or Equation 22 to find the number of punches. Using Equation 11:\newlineT=6+5(2)T = 6 + 5(2)\newlineT=16T = 16

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