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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Leroy, a frequent business traveler, is considering signing up for a hotel rewards program. Hotel Elegance is currently offering 648 points for signing up, plus 24 points for every night Leroy books at the hotel. Alternatively, Hotel Paradiso is offering a sign-up bonus of 412 points, plus 83 points per night. If Leroy books a certain number of nights, the points he earns with either hotel will be the same. How many nights will that be? How many points will he have? If he books ◻ nights, he will earn ◻ points at either hotel.

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineLeroy, a frequent business traveler, is considering signing up for a hotel rewards program. Hotel Elegance is currently offering 648648 points for signing up, plus 2424 points for every night Leroy books at the hotel. Alternatively, Hotel Paradiso is offering a sign-up bonus of 412412 points, plus 8383 points per night. If Leroy books a certain number of nights, the points he earns with either hotel will be the same. How many nights will that be? How many points will he have?\newlineIf he books □\square nights, he will earn □\square points at either hotel.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineLeroy, a frequent business traveler, is considering signing up for a hotel rewards program. Hotel Elegance is currently offering 648648 points for signing up, plus 2424 points for every night Leroy books at the hotel. Alternatively, Hotel Paradiso is offering a sign-up bonus of 412412 points, plus 8383 points per night. If Leroy books a certain number of nights, the points he earns with either hotel will be the same. How many nights will that be? How many points will he have?\newlineIf he books □\square nights, he will earn □\square points at either hotel.
  1. Define Variables: Let's define the variables:\newlineLet xx be the number of nights Leroy books.\newlineLet yy be the total points Leroy earns.\newlineWe can write two equations to represent the points Leroy earns from each hotel:\newlineFor Hotel Elegance: y=648+24xy = 648 + 24x\newlineFor Hotel Paradiso: y=412+83xy = 412 + 83x\newlineWe want to find the value of xx for which the total points yy from both hotels are the same.
  2. Write Equations: To solve the system using substitution, we set the two expressions for yy equal to each other because we are looking for the number of nights where Leroy would earn the same points from both hotels: 648+24x=412+83x648 + 24x = 412 + 83x
  3. Substitute and Solve: Now, we solve for xx: Subtract 24x24x from both sides: 648+24x−24x=412+83x−24x648 + 24x - 24x = 412 + 83x - 24x 648=412+59x648 = 412 + 59x Subtract 412412 from both sides: 648−412=59x648 - 412 = 59x 236=59x236 = 59x Divide both sides by 5959: 23659=x\frac{236}{59} = x 4=x4 = x
  4. Calculate Total Points: Leroy needs to book 44 nights to earn the same number of points at either hotel. Now we need to calculate the total points he will earn for those 44 nights.\newlineWe can substitute x=4x = 4 into either of the original equations. Let's use the equation for Hotel Elegance:\newliney=648+24(4)y = 648 + 24(4)
  5. Calculate Total Points: Leroy needs to book 44 nights to earn the same number of points at either hotel. Now we need to calculate the total points he will earn for those 44 nights.\newlineWe can substitute x=4x = 4 into either of the original equations. Let's use the equation for Hotel Elegance:\newliney=648+24(4)y = 648 + 24(4)Calculate the total points:\newliney=648+96y = 648 + 96\newliney=744y = 744

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