Question

write a system of equations to describe the situation below, solve using any method, and fill in the blanks. jim, a frequent business traveler, is considering signing up for a hotel rewards program. hotel elegance is currently offering 272 points for signing up, plus 94 points for every night jim books at the hotel. alternatively, hotel paradiso is offering a sign - up bonus of 407 points, plus 89 points per night. if jim books a certain number of nights, the points he earns with either hotel will be the same. how many points will he have? how many nights will that be? jim will have points at either hotel by booking nights. submit

Explanation:

Step1: Set up the equations

Let \(x\) be the number of nights and \(y\) be the total points.
For Hotel Elegance: \(y = 272+94x\)
For Hotel Paradiso: \(y = 407 + 89x\)

Step2: Solve the system of equations

Since \(y=y\), we set \(272+94x=407 + 89x\)
Subtract \(89x\) from both sides: \(272+94x-89x=407+89x - 89x\)
\(272 + 5x=407\)
Subtract \(272\) from both sides: \(5x=407 - 272\)
\(5x=135\)
Divide both sides by \(5\): \(x=\frac{135}{5}=27\)

Step3: Find the value of \(y\)

Substitute \(x = 27\) into \(y = 272+94x\)
\(y=272+94\times27\)
\(y=272 + 2538\)
\(y=2810\)

Answer:

Jim will have \(2810\) points at either hotel by booking \(27\) nights.