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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineTwo sisters have decided to watch all the episodes of a popular TV show on DVD. Eva has already watched 22 episodes and will continue to watch the show at a rate of 22 episodes per day. Olivia, who hasn't yet seen the show, will start watching 44 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?\newlineIn _\_ days, the sisters will each have watched _\_ episodes of the show.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineTwo sisters have decided to watch all the episodes of a popular TV show on DVD. Eva has already watched 22 episodes and will continue to watch the show at a rate of 22 episodes per day. Olivia, who hasn't yet seen the show, will start watching 44 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?\newlineIn _\_ days, the sisters will each have watched _\_ episodes of the show.
  1. Define Variables: Let's define the variables first. Let xx represent the number of days, and yy represent the number of episodes watched.\newlineFor Eva:\newlineNumber of episodes already watched: 22\newlineWatching rate of episodes per day: 22\newlineThe equation based on the given information will be:\newlineEpisodes watched = watching rate ∗* days + number of episodes already watched\newliney=2x+2y = 2x + 2\newlineFor Eva, the equation is: y=2x+2y = 2x + 2
  2. Eva's Information: For Olivia:\newlineNumber of episodes already watched: 00 (since she hasn't started yet)\newlineWatching rate of episodes per day: 44\newlineThe equation based on the given information will be:\newlineEpisodes watched = watching rate ×\times days + number of episodes already watched\newliney=4x+0y = 4x + 0\newlineFor Olivia, the equation is: y=4xy = 4x
  3. Olivia's Information: Now we have a system of equations:\newliney=2x+2y = 2x + 2\newliney=4xy = 4x\newlineTo find the point where they have watched the same number of episodes, we set the two equations equal to each other:\newline2x+2=4x2x + 2 = 4x
  4. System of Equations: We will solve for xx:
    Subtract 2x2x from both sides of the equation to get:
    2x+2−2x=4x−2x2x + 2 - 2x = 4x - 2x
    2=2x2 = 2x
    Now, divide both sides by 22 to isolate xx:
    22=2x2\frac{2}{2} = \frac{2x}{2}
    1=x1 = x
    So, x=1x = 1
  5. Solve for x: We found x=1x = 1. Now we need to find the value of yy by substituting xx into one of the original equations. Let's use Eva's equation:\newlineSubstitute 11 for xx in y=2x+2y = 2x + 2:\newlineyy\newline=2(1)+2= 2(1) + 2\newline=2+2= 2 + 2\newline=4= 4\newlineSo, yy00
  6. Substitute xx into Equation: We found:\newlinex=1x = 1\newliney=4y = 4\newlineThis means that in 11 day, the sisters will each have watched 44 episodes of the show.

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