Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Two sisters have decided to watch all the episodes of a popular TV show on DVD. Jenny has already watched $8$ episodes and will continue to watch the show at a rate of $1$ episode per day. Olivia, who hasn't yet seen the show, will start watching $2$ episodes per day. Once the two sisters get to the point where they have watched the same number of episodes, they plan to finish the series together. How many episodes will each sister have watched at that point? How long will that take? $\newline$ The two sisters will each have watched $\_$ episodes of the show in $\_$ days.

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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

Two sisters have decided to watch all the episodes of a popular TV show on DVD. Jenny has already watched

8

episodes and will continue to watch the show at a rate of

1

episode per day. Olivia, who hasn't yet seen the show, will start watching

2

episodes per day. Once the two sisters get to the point where they have watched the same number of episodes, they plan to finish the series together. How many episodes will each sister have watched at that point? How long will that take?

\newline

The two sisters will each have watched

\_

episodes of the show in

\_

days.

Define Variables: Define variables for the number of episodes watched by Jenny and Olivia and the number of days that have passed. $\newline$ Let's let $J$ represent the total number of episodes Jenny has watched and $O$ represent the total number of episodes Olivia has watched. Let $d$ represent the number of days that have passed since Olivia started watching.
Write Jenny's Equation: Write an equation for Jenny's progress. $\newline$ Jenny has already watched $8$ episodes and watches $1$ episode per day. Therefore, her total number of episodes watched as a function of days is: $\newline$ $J = 8 + 1d$
Write Olivia's Equation: Write an equation for Olivia's progress. $\newline$ Olivia starts from $0$ episodes and watches $2$ episodes per day. Therefore, her total number of episodes watched as a function of days is: $\newline$ $O = 2d$
Set Up System of Equations: Set up the system of equations to find when Jenny and Olivia have watched the same number of episodes. $\newline$ Since we are looking for the point where they have watched the same number of episodes, we can set $J$ equal to $O$ : $\newline$ $8 + 1d = 2d$
Solve for $d$ : Solve the equation for $d$ . $\newline$ Subtract $1d$ from both sides to isolate $d$ : $\newline$ $8 + 1d - 1d = 2d - 1d$ $\newline$ $8 = d$
Determine Episodes Watched: Determine the number of episodes each sister has watched. $\newline$ Now that we know it will take $8$ days for them to have watched the same number of episodes, we can find out how many episodes each sister has watched by substituting $d$ back into either $J$ or $O$ . $\newline$ For Jenny: $\newline$ $J = 8 + 1d$ $\newline$ $J = 8 + 1(8)$ $\newline$ $J = 16$ $\newline$ For Olivia: $\newline$ $O = 2d$ $\newline$ $O = 2(8)$ $\newline$ $O = 16$
Answer Prompt: Answer the question prompt. $\newline$ The two sisters will each have watched $16$ episodes of the show in $8$ days.