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Two classmates have decided to read all the volumes in a popular series of books. Amy has already read $10$ volumes and will continue to read new ones at a rate of $1$ volume per week. Beth, who hasn't started reading the series yet, will read $3$ volumes per week. At some point, Beth will catch up with Amy and they will be reading the same book. How many volumes will each girl have read by then? $\newline$ Write a system of equations, graph them, and type the solution. $\newline$ ____ volumes $\newline$

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Write and solve equations for proportional relationships

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Q. Two classmates have decided to read all the volumes in a popular series of books. Amy has already read

10

volumes and will continue to read new ones at a rate of

1

volume per week. Beth, who hasn't started reading the series yet, will read

3

volumes per week. At some point, Beth will catch up with Amy and they will be reading the same book. How many volumes will each girl have read by then?

\newline

Write a system of equations, graph them, and type the solution.

\newline

____ volumes

\newline

Define number of weeks: Let's define the number of weeks as $x$ . Amy starts with $10$ volumes and reads $1$ more each week, so her equation is $A = 10 + x$ . Beth starts with $0$ volumes and reads $3$ each week, so her equation is $B = 3x$ .
Set equations equal: To find when Beth catches up to Amy, set the equations equal: $10 + x = 3x$ .
Solve for x: Solve for $x$ by subtracting $x$ from both sides: $10 = 2x$ .
Isolate x: Divide both sides by $2$ to isolate $x$ : $x = 5$ .
Plug in x: Plug $x = 5$ back into either equation to find the number of volumes each has read. Using Amy's equation: $A = 10 + 5 = 15$ .

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