Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Kevin has punch cards for his favorite tea house and his favorite coffee shop. He currently has $6$ punches on the tea punch card and $9$ punches on the coffee punch card. Given his regular routine, he consistently earns $6$ new punches per week on the tea punch card and $3$ 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? $\newline$ Kevin will have $\_$ punches on each card in $\_$ weeks.

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Solve a system of equations using substitution: word problems

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

\newline

Kevin has punch cards for his favorite tea house and his favorite coffee shop. He currently has

6

punches on the tea punch card and

9

punches on the coffee punch card. Given his regular routine, he consistently earns

6

new punches per week on the tea punch card and

3

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?

\newline

Kevin will have

\_

punches on each card in

\_

weeks.

Define Variables: Define the variables for the system of equations. $\newline$ Let's let $T$ represent the total number of punches on the tea punch card and $C$ represent the total number of punches on the coffee punch card. We know that Kevin starts with $6$ punches on the tea card and $9$ on the coffee card. He earns $6$ punches per week on the tea card and $3$ punches per week on the coffee card. We want to find out after how many weeks, $w$ , he will have the same number of punches on both cards.
Write Equations: Write the system of equations based on the information given. $\newline$ The first equation will represent the total number of punches on the tea card after $w$ weeks, which is the initial $6$ punches plus $6$ times the number of weeks: $\newline$ $T = 6 + 6w$ $\newline$ The second equation will represent the total number of punches on the coffee card after $w$ weeks, which is the initial $9$ punches plus $3$ times the number of weeks: $\newline$ $C = 9 + 3w$ $\newline$ We want to find the point where $T$ equals $C$ .
Use Substitution: Use substitution to solve the system of equations. $\newline$ Since we are looking for when $T$ equals $C$ , we can set the two equations equal to each other: $\newline$ $6 + 6w = 9 + 3w$ $\newline$ Now we will solve for $w$ .
Solve for w: Solve for w. $\newline$ Subtract $3w$ from both sides of the equation to get the $w$ terms on one side: $\newline$ $6 + 6w - 3w = 9 + 3w - 3w$ $\newline$ This simplifies to: $\newline$ $6 + 3w = 9$ $\newline$ Now, subtract $6$ from both sides to isolate the $w$ term: $\newline$ $3w = 9 - 6$ $\newline$ $3w = 3$ $\newline$ Divide both sides by $3$ to solve for $w$ : $\newline$ $w$ $0$ $\newline$ $w$ $1$
Determine Punches: Determine the number of punches on each card after $w$ weeks. $\newline$ Now that we know $w = 1$ , we can substitute this value back into either of the original equations to find the number of punches. We'll use the first equation: $\newline$ $T = 6 + 6w$ $\newline$ $T = 6 + 6(1)$ $\newline$ $T = 6 + 6$ $\newline$ $T = 12$ $\newline$ Since $T$ equals $C$ when $w = 1$ , Kevin will have $12$ punches on each card.