Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Question

Philip has punch cards for his favorite tea house and his favorite coffee shop. He currently has 7 punches on the tea punch card and 11 punches on the coffee punch card. Given his regular routine, he consistently earns 2 new punches per week on the tea punch card and 1 on the coffee punch card. Before too long, Philip will have the same number of punches on each card. How many punches will Philip have on each card? How long will that take?

Philip will have
punches on each card in
weeks.

Answer 1

Let's define the variables for the problem:

• Let \( T \) be the number of punches on the tea card after \( t \) weeks.
• Let \( C \) be the number of punches on the coffee card after \( t \) weeks.

Initially, Philip has:

• \( T = 7 \) (initial punches on tea card)
• \( C = 11 \) (initial punches on coffee card)

The rate of earning punches is as follows:

• Philip earns 2 punches per week on the tea card: \( T = 7 + 2t \)
• Philip earns 1 punch per week on the coffee card: \( C = 11 + t \)

Now, we want to find when the number of punches on both cards will be the same, so we set up the equation:

\[ 7 + 2t = 11 + t \]

Now, we will solve for \( t \):

1. Subtract \( t \) from both sides:

\[ 7 + 2t - t = 11 \]

2. Combine like terms:

\[ 7 + t = 11 \]

3. Subtract 7 from both sides:

\[ t = 11 - 7 \]

4. Thus,

\[ t = 4 \]

Now, substituting \( t = 4 \) back into the equations for \( T \) and \( C \) to find the number of punches on each card:

For the tea card:

\[ T = 7 + 2(4) = 7 + 8 = 15 \]

For the coffee card:

\[ C = 11 + 4 = 15 \]

So, the final result is:

Philip will have 15 punches on each card in 4 weeks.

To summarize:

Philip will have 15 punches on each card in 4 weeks.