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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Khalil has punch cards for his favorite tea house and his favorite coffee shop. He currently has $2$ punches on the tea punch card and $5$ punches on the coffee punch card. Given his regular routine, he consistently earns $5$ new punches per week on the tea punch card and $2$ on the coffee punch card. Before too long, Khalil will have the same number of punches on each card. How many punches will Khalil have on each card? How long will that take? $\newline$ Khalil will have $\_$ punches on each card in $\_$ weeks.

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Solve a system of equations using substitution: word problems

Full solution

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

\newline

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

2

punches on the tea punch card and

5

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

5

new punches per week on the tea punch card and

2

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

\newline

Khalil will have

\_

punches on each card in

\_

weeks.

Define equations for punch cards: Step $1$ : Define the equations for Khalil's punch cards. $\newline$ For the tea punch card, Khalil starts with $2$ punches and earns $5$ punches per week. The equation representing the total punches on the tea card over time is: $\newline$ $y = 5x + 2$ $\newline$ For the coffee punch card, Khalil starts with $5$ punches and earns $2$ punches per week. The equation representing the total punches on the coffee card over time is: $\newline$ $y = 2x + 5$
Set equations equal to solve: Step $2$ : Set the equations equal to solve for x using substitution. $\newline$ $5x + 2 = 2x + 5$ $\newline$ Solving for x: $\newline$ $5x - 2x = 5 - 2$ $\newline$ $3x = 3$ $\newline$ $x = 1$
Substitute x to find y: Step $3$ : Substitute x = $1$ back into either equation to find y. $\newline$ Using the tea punch card equation: $\newline$ $y = 5(1) + 2$ $\newline$ $y = 7$

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