A boutique hotel offers two types of rooms: Executive Suites and Deluxe Suites. During a c… — GMAT

A boutique hotel offers two types of rooms: Executive Suites and Deluxe Suites. During a convention weekend, booking $12$ Executive Suites and $18$ Deluxe Suites generated total revenue of $\$7{}800$ . The following weekend, booking $15$ Executive Suites and $10$ Deluxe Suites generated total revenue of $\$7{}250$ . If the nightly rate for each type of suite remained constant, what was the nightly rental rate, in dollars, of an Executive Suite?

Answer: 350 dollars

Answer

The nightly rental rate of an Executive Suite was 350 dollars.

Formulating the revenue statements as linear equations gives

12x + 18y = 7,800

and

15x + 10y = 7,250

. Simplifying these by dividing by

6

and

5

yields

2x + 3y = 1,300

and

3x + 2y = 1,450

. Multiplying the first simplified equation by

2

(

4x + 6y = 2,600

) and the second by

3

(

9x + 6y = 4,350

) aligns the

y

-coefficients. Subtracting the equations leads to

5x = 1,750

, giving

x = 350

Step-by-Step Solution

Define variables and construct the system of equations.

12x + 18y = 7,800

and

15x + 10y = 7,250

, where

x

represents the Executive Suite rate and

y

represents the Deluxe Suite rate.

Translating word problem information into algebraic equations.

Simplify the system equations.

Dividing the first equation by

6

gives

2x + 3y = 1,300

. Dividing the second equation by

5

gives

3x + 2y = 1,450

Reducing the coefficients minimizes computation steps and errors.

Eliminate variable

y

to solve for

x

Multiply

2x + 3y = 1,300

2

to get

4x + 6y = 2,600

. Multiply

3x + 2y = 1,450

3

to get

9x + 6y = 4,350

. Subtracting gives

5x = 1,750

, so

x = 350

Eliminating

y

yields the requested value of

x

directly.

Key Concept

Solving Systems of Two Linear Equations via Elimination

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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