Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Emmy and Danielle are making presentations for a class project. Emmy's slideshow starts with a verbal introduction that is $20$ seconds long, and then each slide is left up for $9$ seconds. Danielle leaves each slide onscreen for $10$ seconds, and her introduction lasts $5$ seconds. Emmy and Danielle notice that their presentations have both the same number of slides and the same duration. How many slides are in each presentation? How long is each presentation? $\newline$ Each presentation has $\_$ slides and is $\_$ seconds long.

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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

Emmy and Danielle are making presentations for a class project. Emmy's slideshow starts with a verbal introduction that is

20

seconds long, and then each slide is left up for

9

seconds. Danielle leaves each slide onscreen for

10

seconds, and her introduction lasts

5

seconds. Emmy and Danielle notice that their presentations have both the same number of slides and the same duration. How many slides are in each presentation? How long is each presentation?

\newline

Each presentation has

\_

slides and is

\_

seconds long.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of slides in each presentation. $\newline$ Let $y$ be the total duration of each presentation in seconds. $\newline$ We can write two equations based on the information given: $\newline$ For Emmy's presentation: The total duration is the time for the introduction plus the time for all the slides. $\newline$ $y = 20 + 9x$ $\newline$ For Danielle's presentation: The total duration is also the time for the introduction plus the time for all the slides. $\newline$ $y = 5 + 10x$ $\newline$ Now we can rephrase the question_prompt: How many slides are in each presentation and how long is each presentation?
Write Equations: We have two equations: $\newline$ $1$ ) $y = 20 + 9x$ (Emmy's presentation) $\newline$ $2$ ) $y = 5 + 10x$ (Danielle's presentation) $\newline$ We can use substitution to solve for $x$ by setting the two equations equal to each other since $y$ is the same for both presentations. $\newline$ $20 + 9x = 5 + 10x$
Use Substitution: Now we solve for $x$ : $20 + 9x = 5 + 10x$ $9x - 10x = 5 - 20$ $-x = -15$ $x = 15$ So, there are $15$ slides in each presentation.
Solve for x: Now we need to find the total duration $y$ for each presentation. We can substitute $x$ back into either of the original equations. Let's use Emmy's equation: $\newline$ $y = 20 + 9x$ $\newline$ $y = 20 + 9(15)$ $\newline$ $y = 20 + 135$ $\newline$ $y = 155$ $\newline$ So, each presentation is $155$ seconds long.