Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Josiah has punch cards for his favorite tea house and his favorite coffee shop. He currently has 2 punches on the tea punch card and 11 punches on the coffee punch card. Given his regular routine, he consistently earns 7 new punches per week on the tea punch card and 4 on the coffee punch card. Before too long, Josiah will have the same number of punches on each card. How long will that take? How many punches will Josiah have on each card?

Answer 1

Let x be the number of weeks; therefore, the two linear equations are

`\begin{gathered} T=2+7x \\ \text{and} \\ C=11+4x \end{gathered}`

Where T stands for the punches on the tea punch card and C for the punches on the coffee punch card.

Solving by substitution. Set T=C, then

`\begin{gathered} T=C \\ \Rightarrow2+7x=11+4x \\ \Rightarrow3x=9 \\ \Rightarrow x=(9)/(3)=3 \\ \Rightarrow x=3 \end{gathered}`

Thus, substitute the value of x=3 into the first equation,

`\begin{gathered} x=3 \\ \Rightarrow T=2+7\cdot3=23 \\ \Rightarrow T=23 \end{gathered}`

Thus, after 3 weeks, Josiah will have the same number of punches on each card, and he will have 23 punches on each card.