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Chapter 8: Problem 78

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. The Executive Inn rents a double room for \(\$ 20\) more per night than a single. One night the motel took in \(\$ 3949\) by renting 15 doubles and 26 singles. What is the rental price for each type of room?

Short Answer

Expert verified
The rental price is \$89 for a single room and \$109 for a double room.

Step by step solution

01

Define the Variables

Let the price of renting a single room per night be denoted as \(S\). Let the price of renting a double room per night be denoted as \(D\).
02

Set Up the Equations

According to the problem, the cost for a double room is \(D = S + 20\). Additionally, we know that the total revenue from renting 15 doubles and 26 singles is \$3949. This gives us the equation: \[15D + 26S = 3949\].
03

Substitute the First Equation into the Second

We substitute \(D = S + 20\) into \[15D + 26S = 3949\]. This gives: \[15(S + 20) + 26S = 3949\].
04

Simplify the Equation

Expand the equation: \[15S + 300 + 26S = 3949\]. Combine like terms: \[41S + 300 = 3949\]. Subtract 300 from both sides: \[41S = 3649\].
05

Solve for S

Divide both sides by 41: \[S = \frac{3649}{41} = 89\]. Thus, the price of a single room is \$89.
06

Solve for D

Using the equation \(D = S + 20\), substitute \(S = 89\): \[D = 89 + 20 = 109\]. Thus, the price of a double room is \$109.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

system of linear equations
A system of linear equations consists of two or more equations with the same set of variables. In solving the problem about room rentals, we used two equations and two variables. The first equation relates the price difference between double and single rooms, and the second equation relates the total revenue with the number of rooms rented.
substitution method
The substitution method is a technique to solve systems of equations. In this method, you solve one equation for one variable and then substitute this expression into the other equation. For instance, we found that the cost of a double room (\(D\)) is 20 dollars more than a single room (\(S\)), giving us the equation \( D = S + 20 \). Substituting \( D = S + 20 \) into the total revenue equation helps us solve for \( S \) first, and then \( D \).
dependent and inconsistent systems
A dependent system has infinitely many solutions. This happens when the equations describe the same line. An inconsistent system has no solutions because the lines are parallel and never intersect. In our exercise, since we solved for specific values of \( S \) and \( D \)...

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