For his parents' anniversary party, Lee is considering using one of two venues. A hotel in Clarksville will cost $500 for a reservation, plus $8 per person. A restaurant in the same city will cost $6 per person, in addition to $600 for the reservation. In order to make the best decision, Lee figures out how many attendees it would take to have the venues cost the same amount. How many attendees would that be?

Question

Write a system of equations, graph them, and type the solution.

Answer 1

To determine the number of attendees at which the costs of both venues would be the same, we can set up equations based on the costs provided for each venue.

Cost Equations

1. Hotel in Clarksville:

   • Fixed cost = $500
   • Variable cost per person = $8
   • Total cost for \( x \) attendees: \[ C_h = 500 + 8x \]

2. Restaurant:

   • Fixed cost = $600
   • Variable cost per person = $6
   • Total cost for \( x \) attendees: \[ C_r = 600 + 6x \]

Setting up the System of Equations

To find the number of attendees \( x \) at which both costs are equal, we can set the two cost equations equal to each other: \[ 500 + 8x = 600 + 6x \]

Solving the Equation

1. Subtract \( 6x \) from both sides: \[ 500 + 8x - 6x = 600 \] \[ 500 + 2x = 600 \]

2. Subtract 500 from both sides: \[ 2x = 100 \]

3. Divide by 2: \[ x = 50 \]

Conclusion

The number of attendees at which the costs of the two venues are equal is 50 attendees.

Graphical Representation

To graph the equations, we would plot the cost functions for both venues:

• For the hotel, the line equation: \[ C_h = 8x + 500 \]
• For the restaurant, the line equation: \[ C_r = 6x + 600 \]

Graphing these two equations will show that they intersect at the point \( (50, C) \) where \( C \) is the cost at that number of attendees.

Result

Thus, Lee finds that he would need 50 attendees for both venues to have the same cost.