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. Sophia has already watched $3$ episodes and will continue to watch the show at a rate of $1$ episode per day. Jenny, 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. Sophia has already watched

3

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

1

episode per day. Jenny, 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 $x$ represent the number of days and $y$ represent the number of episodes watched. $\newline$ For Sophia: $\newline$ Number of episodes already watched: $3$ $\newline$ Watching rate of episodes per day: $1$ $\newline$ The equation based on the given information is: $\newline$ Episodes watched = watching rate $\times$ days + number of episodes already watched $\newline$ $y = 1x + 3$ $\newline$ For Sophia, the equation is: $y = x + 3$
Sophia's Equation: For Jenny: $\newline$ 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 is: $\newline$ Episodes watched = watching rate $\times$ days + number of episodes already watched $\newline$ $y = 4x + 0$ $\newline$ For Jenny, the equation is: $y = 4x$
Jenny's Equation: System of equations: $\newline$ $y = x + 3$ $\newline$ $y = 4x$ $\newline$ To find the point where they have watched the same number of episodes, set the two equations equal to each other. $\newline$ $x + 3 = 4x$
System of Equations: Solve for $x$ by isolating the variable. $\newline$ Subtract $x$ from both sides of the equation: $\newline$ $x + 3 - x = 4x - x$ $\newline$ $3 = 3x$ $\newline$ Now, divide both sides by $3$ to solve for $x$ : $\newline$ $\frac{3}{3} = \frac{3x}{3}$ $\newline$ $1 = x$ $\newline$ So, $x = 1$
Solve for $x$ : Find the value of $y$ by substituting $x$ into one of the original equations. $\newline$ Substitute $1$ for $x$ in $y = 4x$ (Jenny's equation): $\newline$ $y = 4(1)$ $\newline$ $y = 4$ $\newline$ So, $y = 4$
Find $y$ : Verify the solution by substituting $x = 1$ into Sophia's equation to ensure it gives the same $y$ value. $\newline$ Substitute $1$ for $x$ in $y = x + 3$ (Sophia's equation): $\newline$ $y = 1 + 3$ $\newline$ $y = 4$ $\newline$ Since this matches the $y$ value found using Jenny's equation, the solution is correct.