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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. Sophia has already watched 33 episodes and will continue to watch the show at a rate of 11 episode per day. Jenny, 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. Sophia has already watched 33 episodes and will continue to watch the show at a rate of 11 episode per day. Jenny, 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 xx represent the number of days and yy represent the number of episodes watched.\newlineFor Sophia:\newlineNumber of episodes already watched: 33\newlineWatching rate of episodes per day: 11\newlineThe equation based on the given information is:\newlineEpisodes watched = watching rate ×\times days + number of episodes already watched\newliney=1x+3y = 1x + 3\newlineFor Sophia, the equation is: y=x+3y = x + 3
  2. Sophia's Equation: For Jenny:\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 is:\newlineEpisodes watched = watching rate ×\times days + number of episodes already watched\newliney=4x+0y = 4x + 0\newlineFor Jenny, the equation is: y=4xy = 4x
  3. Jenny's Equation: System of equations:\newliney=x+3y = x + 3\newliney=4xy = 4x\newlineTo find the point where they have watched the same number of episodes, set the two equations equal to each other.\newlinex+3=4xx + 3 = 4x
  4. System of Equations: Solve for xx by isolating the variable.\newlineSubtract xx from both sides of the equation:\newlinex+3−x=4x−xx + 3 - x = 4x - x\newline3=3x3 = 3x\newlineNow, divide both sides by 33 to solve for xx:\newline33=3x3\frac{3}{3} = \frac{3x}{3}\newline1=x1 = x\newlineSo, x=1x = 1
  5. Solve for xx: Find the value of yy by substituting xx into one of the original equations.\newlineSubstitute 11 for xx in y=4xy = 4x (Jenny's equation):\newliney=4(1)y = 4(1)\newliney=4y = 4\newlineSo, y=4y = 4
  6. Find yy: Verify the solution by substituting x=1x = 1 into Sophia's equation to ensure it gives the same yy value.\newlineSubstitute 11 for xx in y=x+3y = x + 3 (Sophia's equation):\newliney=1+3y = 1 + 3\newliney=4y = 4\newlineSince this matches the yy value found using Jenny's equation, the solution is correct.

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