Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Jeanette has punch cards for her favorite tea house and her favorite coffee shop. She currently has $10$ punches on the tea punch card and $1$ punch on the coffee punch card. Given her regular routine, she consistently earns $2$ new punches per week on the tea punch card and $3$ on the coffee punch card. Before too long, Jeanette will have the same number of punches on each card. How many punches will Jeanette have on each card? How long will that take? $\newline$ Jeanette will have $\_$ punches on each card in $\_$ weeks.

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Solve a system of equations using any method: word problems

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

\newline

Jeanette has punch cards for her favorite tea house and her favorite coffee shop. She currently has

10

punches on the tea punch card and

1

punch on the coffee punch card. Given her regular routine, she consistently earns

2

new punches per week on the tea punch card and

3

on the coffee punch card. Before too long, Jeanette will have the same number of punches on each card. How many punches will Jeanette have on each card? How long will that take?

\newline

Jeanette will have

\_

punches on each card in

\_

weeks.

Define variables: Let's define the variables: Let $x$ be the number of weeks it takes for Jeanette to have the same number of punches on both cards. Let $T$ be the total number of punches on the tea punch card, and $C$ be the total number of punches on the coffee punch card after $x$ weeks.
Write equations: We can write two equations based on the information given. Jeanette starts with $10$ punches on the tea card and earns $2$ punches per week. So, after $x$ weeks, she will have $T = 10 + 2x$ punches on the tea card.
Set equations equal: Similarly, Jeanette starts with $1$ punch on the coffee card and earns $3$ punches per week. So, after $x$ weeks, she will have $C = 1 + 3x$ punches on the coffee card.
Solve for x: We want to find out when $T$ equals $C$ , so we set the two equations equal to each other: $10 + 2x = 1 + 3x$ .
Substitute $x$ into equations: Now, we solve for $x$ . Subtract $2x$ from both sides to get: $10 = 1 + x$ .
Calculate punches after $9$ weeks: Subtract $1$ from both sides to isolate $x$ : $9 = x$ .
Check work: Now that we have $x$ , we can find out how many punches Jeanette will have on each card after $9$ weeks. We substitute $x = 9$ into either $T = 10 + 2x$ or $C = 1 + 3x$ .
Check work: Now that we have $x$ , we can find out how many punches Jeanette will have on each card after $9$ weeks. We substitute $x = 9$ into either $T = 10 + 2x$ or $C = 1 + 3x$ .Substituting $x = 9$ into $T = 10 + 2x$ gives us $T = 10 + 2(9) = 10 + 18 = 28$ .
Check work: Now that we have $x$ , we can find out how many punches Jeanette will have on each card after $9$ weeks. We substitute $x = 9$ into either $T = 10 + 2x$ or $C = 1 + 3x$ .Substituting $x = 9$ into $T = 10 + 2x$ gives us $T = 10 + 2(9) = 10 + 18 = 28$ .To check our work, we substitute $x = 9$ into $C = 1 + 3x$ as well: $9$ $0$ .
Check work: Now that we have $x$ , we can find out how many punches Jeanette will have on each card after $9$ weeks. We substitute $x = 9$ into either $T = 10 + 2x$ or $C = 1 + 3x$ .Substituting $x = 9$ into $T = 10 + 2x$ gives us $T = 10 + 2(9) = 10 + 18 = 28$ .To check our work, we substitute $x = 9$ into $C = 1 + 3x$ as well: $9$ $0$ .Since $9$ $1$ and $9$ $2$ are equal, we have found the correct number of punches on each card after $9$ weeks. Jeanette will have $9$ $4$ punches on each card in $9$ weeks.