Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two sisters have decided to watch all the episodes of a popular TV show on DVD. Eva has already watched $2$ episodes and will continue to watch the show at a rate of $2$ episodes per day. Olivia, who hasn't yet seen the show, will start watching $4$ 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 long will that take? How many episodes will each sister have watched at that point? $\newline$ In $\_$ days, the sisters will each have watched $\_$ episodes of the show.

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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

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

2

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

2

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

4

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 long will that take? How many episodes will each sister have watched at that point?

\newline

\_

days, the sisters will each have watched

\_

episodes of the show.

Define Variables: Let's define the variables first. Let $x$ represent the number of days, and $y$ represent the number of episodes watched. $\newline$ For Eva: $\newline$ Number of episodes already watched: $2$ $\newline$ Watching rate of episodes per day: $2$ $\newline$ The equation based on the given information will be: $\newline$ Episodes watched = watching rate $*$ days + number of episodes already watched $\newline$ $y = 2x + 2$ $\newline$ For Eva, the equation is: $y = 2x + 2$
Eva's Information: For Olivia: $\newline$ Number of episodes already watched: $0$ (since she hasn't started yet) $\newline$ Watching rate of episodes per day: $4$ $\newline$ The equation based on the given information will be: $\newline$ Episodes watched = watching rate $\times$ days + number of episodes already watched $\newline$ $y = 4x + 0$ $\newline$ For Olivia, the equation is: $y = 4x$
Olivia's Information: Now we have a system of equations: $\newline$ $y = 2x + 2$ $\newline$ $y = 4x$ $\newline$ To find the point where they have watched the same number of episodes, we set the two equations equal to each other: $\newline$ $2x + 2 = 4x$
System of Equations: We will solve for $x$ :
Subtract $2x$ from both sides of the equation to get:
$2x + 2 - 2x = 4x - 2x$
$2 = 2x$
Now, divide both sides by $2$ to isolate $x$ :
$\frac{2}{2} = \frac{2x}{2}$
$1 = x$
So, $x = 1$
Solve for x: We found $x = 1$ . Now we need to find the value of $y$ by substituting $x$ into one of the original equations. Let's use Eva's equation: $\newline$ Substitute $1$ for $x$ in $y = 2x + 2$ : $\newline$ $y$ $\newline$ $= 2(1) + 2$ $\newline$ $= 2 + 2$ $\newline$ $= 4$ $\newline$ So, $y$ $0$
Substitute $x$ into Equation: We found: $\newline$ $x = 1$ $\newline$ $y = 4$ $\newline$ This means that in $1$ day, the sisters will each have watched $4$ episodes of the show.