Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ 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? $\newline$ If he books nights, he will earn points at either hotel. $\newline$ Submit

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Grade 8

Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

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?

\newline

If he books nights, he will earn points at either hotel.

\newline

Submit

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of nights Leroy books. $\newline$ Let $y$ be the total points Leroy earns. $\newline$ We can write two equations to represent the points Leroy earns from each hotel: $\newline$ For Hotel Elegance: $y = 648 + 24x$ $\newline$ For Hotel Paradiso: $y = 412 + 83x$ $\newline$ We want to find the value of $x$ for which the total points $y$ from both hotels are the same.
Write Equations: To solve the system using substitution, we set the two expressions for $y$ equal to each other because we are looking for the number of nights where Leroy would earn the same points from both hotels: $648 + 24x = 412 + 83x$
Substitute and Solve: Now, we solve for $x$ : Subtract $24x$ from both sides: $648 + 24x - 24x = 412 + 83x - 24x$ $648 = 412 + 59x$ Subtract $412$ from both sides: $648 - 412 = 59x$ $236 = 59x$ Divide both sides by $59$ : $\frac{236}{59} = x$ $4 = x$
Calculate Total Points: Leroy needs to book $4$ nights to earn the same number of points at either hotel. Now we need to calculate the total points he will earn for those $4$ nights. $\newline$ We can substitute $x = 4$ into either of the original equations. Let's use the equation for Hotel Elegance: $\newline$ $y = 648 + 24(4)$
Calculate Total Points: Leroy needs to book $4$ nights to earn the same number of points at either hotel. Now we need to calculate the total points he will earn for those $4$ nights. $\newline$ We can substitute $x = 4$ into either of the original equations. Let's use the equation for Hotel Elegance: $\newline$ $y = 648 + 24(4)$ Calculate the total points: $\newline$ $y = 648 + 96$ $\newline$ $y = 744$