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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineKevin has punch cards for his favorite tea house and his favorite coffee shop. He currently has 66 punches on the tea punch card and 99 punches on the coffee punch card. Given his regular routine, he consistently earns 66 new punches per week on the tea punch card and 33 on the coffee punch card. Before too long, Kevin will have the same number of punches on each card. How many punches will Kevin have on each card? How long will that take?\newlineKevin 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.\newlineKevin has punch cards for his favorite tea house and his favorite coffee shop. He currently has 66 punches on the tea punch card and 99 punches on the coffee punch card. Given his regular routine, he consistently earns 66 new punches per week on the tea punch card and 33 on the coffee punch card. Before too long, Kevin will have the same number of punches on each card. How many punches will Kevin have on each card? How long will that take?\newlineKevin will have _\_ punches on each card in _\_ weeks.
  1. Define Variables: Define the variables for the system of equations.\newlineLet's let TT represent the total number of punches on the tea punch card and CC represent the total number of punches on the coffee punch card. We know that Kevin starts with 66 punches on the tea card and 99 on the coffee card. He earns 66 punches per week on the tea card and 33 punches per week on the coffee card. We want to find out after how many weeks, ww, he will have the same number of punches on both cards.
  2. Write Equations: Write the system of equations based on the information given.\newlineThe first equation will represent the total number of punches on the tea card after ww weeks, which is the initial 66 punches plus 66 times the number of weeks:\newlineT=6+6wT = 6 + 6w\newlineThe second equation will represent the total number of punches on the coffee card after ww weeks, which is the initial 99 punches plus 33 times the number of weeks:\newlineC=9+3wC = 9 + 3w\newlineWe want to find the point where TT equals CC.
  3. Use Substitution: Use substitution to solve the system of equations.\newlineSince we are looking for when TT equals CC, we can set the two equations equal to each other:\newline6+6w=9+3w6 + 6w = 9 + 3w\newlineNow we will solve for ww.
  4. Solve for w: Solve for w.\newlineSubtract 3w3w from both sides of the equation to get the ww terms on one side:\newline6+6w−3w=9+3w−3w6 + 6w - 3w = 9 + 3w - 3w\newlineThis simplifies to:\newline6+3w=96 + 3w = 9\newlineNow, subtract 66 from both sides to isolate the ww term:\newline3w=9−63w = 9 - 6\newline3w=33w = 3\newlineDivide both sides by 33 to solve for ww:\newlineww00\newlineww11
  5. Determine Punches: Determine the number of punches on each card after ww weeks.\newlineNow that we know w=1w = 1, we can substitute this value back into either of the original equations to find the number of punches. We'll use the first equation:\newlineT=6+6wT = 6 + 6w\newlineT=6+6(1)T = 6 + 6(1)\newlineT=6+6T = 6 + 6\newlineT=12T = 12\newlineSince TT equals CC when w=1w = 1, Kevin will have 1212 punches on each card.

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