Question

sophia, a frequent business traveler, is considering signing up for a hotel rewards program. hotel elegance is currently offering 331 points for signing up, plus 18 points for every night sophia books at the hotel. alternatively, hotel paradiso is offering a sign - up bonus of 333 points, plus 16 points per night. if sophia books a certain number of nights, the points she earns with either hotel will be the same. how many points will she have? how many nights will that be? sophia will have points at either hotel by booking nights.

Explanation:

Step1: Set up equations

Let \(n\) be the number of nights.
For Hotel Elegance: \(y = 331+18n\)
For Hotel Paradiso: \(y = 333 + 16n\)

Step2: Set equations equal

Set \(331+18n=333 + 16n\)
Subtract \(16n\) from both sides: \(331 + 18n-16n=333+16n - 16n\)
\(331+2n=333\)
Subtract \(331\) from both sides: \(2n=333 - 331\)
\(2n=2\)
Divide both sides by \(2\): \(n = 1\)

Step3: Find points

Substitute \(n = 1\) into either equation.
Using Hotel Elegance: \(y=331+18\times1=331 + 18=349\)

Answer:

Sophia will have \(349\) points at either hotel by booking \(1\) night.