Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ For her parents' anniversary party, Maddie is considering using one of two venues. A hotel in Springfield will cost $\$269$ for a reservation, plus $\$28$ per person. A restaurant in the same city will cost $\$30$ per person, in addition to $\$267$ for the reservation. In order to make the best decision, Maddie figures out how many attendees it would take to have the venues cost the same amount. How many attendees would that be? What would the total cost be? $\newline$ If the party had _____ attendees, the cost at each venue would be $\$\_\_\_\_\_$ .

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Algebra 1

Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

For her parents' anniversary party, Maddie is considering using one of two venues. A hotel in Springfield will cost

\$269

for a reservation, plus

\$28

per person. A restaurant in the same city will cost

\$30

per person, in addition to

\$267

for the reservation. In order to make the best decision, Maddie figures out how many attendees it would take to have the venues cost the same amount. How many attendees would that be? What would the total cost be?

\newline

If the party had _____ attendees, the cost at each venue would be

\$\_\_\_\_\_

Define variables: Let's define the variables. Let $x$ be the number of attendees, and let $C$ be the total cost for either venue.
Write hotel equation: Write the equation for the hotel in Springfield. The cost is $\$269$ for the reservation plus $\$28$ per person. $C = 269 + 28x$
Write restaurant equation: Write the equation for the restaurant in the same city. The cost is $\$267$ for the reservation plus $\$30$ per person. $\newline$ $C = 267 + 30x$
Set equations equal: Set the two equations equal to each other to find the number of attendees for which the cost is the same. $269 + 28x = 267 + 30x$
Solve for x: Solve for x by subtracting $28x$ from both sides of the equation. $\newline$ $269 + 28x - 28x = 267 + 30x - 28x$ $\newline$ $269 = 267 + 2x$
Isolate variable $x$ : Subtract $267$ from both sides to isolate the variable $x$ . $\newline$ $269 - 267 = 267 + 2x - 267$ $\newline$ $2 = 2x$
Divide to solve $x$ : Divide both sides by $2$ to solve for $x$ . $\frac{2}{2} = \frac{2x}{2}$ $1 = x$
Calculate total cost: Now that we have the number of attendees $x$ , we can calculate the total cost $C$ for either venue. $\newline$ $C = 269 + 28(1) = 269 + 28 = 297$ $\newline$ $C = 267 + 30(1) = 267 + 30 = 297$