Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ For his parents' anniversary party, Clarence is considering using one of two venues. A hotel in Millersburg will cost $\$736$ for a reservation, plus $\$26$ per person. A restaurant in the same city will cost $\$31$ per person, in addition to $\$616$ for the reservation. In order to make the best decision, Clarence figures out how many attendees it would take to have the venues cost the same amount. What would the total cost be? How many attendees would that be? $\newline$ The cost at each venue would be $\$$ _ if the party had _ attendees.

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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

For his parents' anniversary party, Clarence is considering using one of two venues. A hotel in Millersburg will cost

\$736

for a reservation, plus

\$26

per person. A restaurant in the same city will cost

\$31

per person, in addition to

\$616

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

\newline

The cost at each venue would be

\$

_____ if the party had _____ attendees.

Set Up Equations: Let $x$ represent the number of attendees and $y$ represent the total cost for the venue. We need to set up two equations, one for each venue. $\newline$ For the hotel in Millersburg: $\newline$ Cost of reservation: $\$736$ $\newline$ Cost per person: $\$26$ $\newline$ The equation based on the given information is: $\newline$ Total cost $y$ = cost per person $*$ number of attendees + cost of reservation $\newline$ $y = 26x + 736$ $\newline$ For the hotel, the equation is: $y = 26x + 736$
Hotel Information: For the restaurant in Millersburg: $\newline$ Cost of reservation: \$$616$ $\newline$Cost per person: \$$31$ $\newline$The equation based on the given information is: $\newline$Total cost $y = 31x + 616$ $\newline$For the restaurant, the equation is: $y = 31x + 616$
Restaurant Information: Now we have a system of equations: $\newline$$y = 26x + 736$ $\newline$$y = 31x + 616$ $\newline$To find the number of attendees where the cost is the same, we set the two equations equal to each other. $\newline$$26x + 736 = 31x + 616$
Solve for x: We need to solve for x. $\newline$First, we'll subtract $26x$ from both sides of the equation to get the x terms on one side: $\newline$$26x + 736 - 26x = 31x + 616 - 26x$ $\newline$This simplifies to: $\newline$$736 = 5x + 616$
Isolate x Term: Next, we'll subtract $616$ from both sides to isolate the x term: $\newline$$736 - 616 = 5x + 616 - 616$$\newline$This simplifies to: $\newline$$120 = 5x$
Divide by $5$: Now we'll divide both sides by $5$ to solve for $x$: $\newline$$\frac{120}{5} = \frac{5x}{5}$ $\newline$This gives us: $\newline$$x = 24$ $\newline$So, the number of attendees needed for the cost to be the same at both venues is $24$.
Substitute $x$: We found $x = 24$. Now we need to find the total cost $y$ by substituting $x$ into one of the original equations. We can use either equation, so we'll use the hotel's equation: $\newline$$y = 26x + 736$ $\newline$Substitute $24$ for $x$: $\newline$$y = 26(24) + 736$
Calculate Total Cost: Calculate the total cost: $\newline$$y = 26 \times 24 + 736$ $\newline$$y = 624 + 736$ $\newline$$y = 1360$ $\newline$So, the total cost at each venue would be $\$1360$ if the party had $24$ attendees.