Question

write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. ben, a frequent business traveler, is considering signing up for a hotel rewards program. in the blanks. ben, a frequent business traveler, is considering signing up for a hotel rewards program. hotel elegance is currently offering 520 points for signing up, plus 72 points for every night ben books at the hotel. alternatively, hotel paradiso is offering a sign - up bonus of 310 points plus 74 points per night. if ben 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? ben will have _ points at either hotel by booking _ nights.

Explanation:

Step1: Define variables

Let $x$ be the number of nights. Let $y$ be the total number of points.
For Hotel Elegance: $y_1 = 220+72x$. For Hotel Paradiso: $y_2=310 + 74x$.

Step2: Set the two - equations equal to each other to find when the points are the same

Since we want to find when $y_1=y_2$, we set up the equation $220 + 72x=310+74x$.

Step3: Solve the equation for $x$

Subtract $72x$ from both sides: $220=310 + 74x-72x$. So, $220=310 + 2x$.
Then subtract 310 from both sides: $220 - 310=2x$, which gives $- 90=2x$.
Divide both sides by 2: $x=-45$ is incorrect. We made a mistake above. Let's start over.
The correct equations should be: For Hotel Elegance: $y = 220+72x$. For Hotel Paradiso: $y=310+74x$.
Set $220 + 72x=310+74x$.
Subtract $72x$ from both sides: $220=310+2x$.
Subtract 310 from both sides: $220 - 310=2x$, so $-90 = 2x$, and $x = 45$.
Substitute $x = 45$ into the first equation $y=220+72\times45=220 + 3240=3460$.

Answer:

3460 points, 45 nights