Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Brian and Eve are making presentations for a class project. Brian's slideshow starts with a verbal introduction that is $13$ seconds long, and then each slide is left up for $3$ seconds. Eve leaves each slide onscreen for $4$ seconds, and her introduction lasts $11$ seconds. Brian and Eve 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

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

13

seconds long, and then each slide is left up for

3

seconds. Eve leaves each slide onscreen for

4

seconds, and her introduction lasts

11

seconds. Brian and Eve 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. $\newline$ Now we can write two equations based on the information given: $\newline$ For Brian's presentation: $y = 13 + 3x$ ( $13$ seconds for the introduction and $3$ seconds for each slide) $\newline$ For Eve's presentation: $y = 11 + 4x$ ( $11$ seconds for the introduction and $4$ seconds for each slide) $\newline$ Since both presentations have the same number of slides and the same duration, we can set the two equations equal to each other: $\newline$ $13 + 3x = 11 + 4x$
Write Equations: Now we solve for $x$ : Subtract $3x$ from both sides: $13 = 11 + x$ Subtract $11$ from both sides: $2 = x$ So, $x = 2$ , which means there are $2$ slides in each presentation.
Set Equations Equal: Next, we substitute $x$ back into one of the original equations to find $y$ . We can use Brian's equation: $\newline$ $y = 13 + 3x$ $\newline$ $y = 13 + 3(2)$ $\newline$ $y = 13 + 6$ $\newline$ $y = 19$ $\newline$ So, the total duration of each presentation is $19$ seconds.