Problem 77 - College Algebra Essentials

Question

A hotel has 200 rooms. Those with kitchen facilities rent for \(\$ 100\) per night and those without kitchen facilities rent for \(\$ 80\) per night. On a night when the hotel was completely occupied, revenues were \(\$ 17,000 .\) How many of each type of room does the hotel have?

Step-by-Step Solution

Answer

There are 50 rooms with kitchen facilities and 150 rooms without kitchen facilities.

Step 1: Definition of Unknowns

Let's denote the number of rooms with kitchen facilities as \(x\) and the number of rooms without kitchen facilities as \(y\).

Step 2: Set up the Equations

Based on the conditions given in the problem, we can set up the following system of equations: Equation 1 (from the total number of rooms): \(x + y = 200\)Equation 2 (from the total revenues): \(100x + 80y = 17000\)

Step 3: Solve the System of Equations

To make the system much easier to solve, we can first multiply the first equation by 80: \(80x + 80y = 16000\). Then we subtract this new equation from the second equation: \(20x = 1000\). Solving for \(x\), we find that \(x = 50\). Then, we substitute \(x = 50\) into the first equation to solve for \(y\), which yields \(y = 150\).

Step 4: Interpret the Result

The solutions \(x = 50\) and \(y = 150\) mean that the hotel has 50 rooms with kitchen facilities and 150 rooms without kitchen facilities.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics when we discuss systems of equations. They represent relationships involving constant terms and products of unknown variables raised to the power of one. In this context, a linear equation creates a straight line when graphed on a coordinate plane. This straight-line relationship allows us to predict values and understand connections between different variables more easily.

In our exercise concerning hotel rooms, the linear equations are formulated based on given conditions. For instance, we know from the problem:

• The total number of rooms is a constant. This gives us the equation: \( x + y = 200 \), where \( x \) and \( y \) are variables representing the number of different types of rooms.
• The total revenue is another constraint: \( 100x + 80y = 17000 \), where each part represents revenue contributions from each type of room.

These equations form a system that we can solve to determine the value of unknown variables. Each line represents a constraint that the solution must satisfy simultaneously.

Unknown Variables

Unknown variables are symbols in an equation that stand for quantities we need to determine. In many problem-solving situations, like the one we have here, defining these variables is the starting point for equation setup.

• In our example, \( x \) represents the number of hotel rooms with kitchen facilities.
• Similarly, \( y \) symbolizes the rooms without kitchen facilities.

This process of defining unknowns is crucial. It frames the problem clearly, turning vague information into something mathematical and solvable.

For effective problem-solving, it is essential to label each unknown variable meaningfully. This ensures that equations are both accurate and easier to comprehend. This step not only simplifies the equations but also helps us make logical deductions from given information and come to a meaningful outcome.

Problem Solving

Problem-solving involves breaking down a problem into smaller, manageable parts and then applying logical reasoning and mathematical tools to find the solution. In exercises involving systems of linear equations, the main goal is to determine the unknown variables by using all the given information correctly.

Here's a strategic breakdown:

• **Define Goals**: Understand what you need to find – for instance, the number of each type of room.
• **Set Up Equations**: Translate the problem into mathematical expressions. In our example, this involves two conditions resulting in two equations.
• **Systematic Solution Methods**: Select an approach like substitution or elimination to solve the system of equations. In this case, manipulating equations to isolate a variable is key.
• **Interpretation**: Translate mathematical answers back to a real-world context, ensuring solutions are reasonable and meet the original problem's conditions.

When approached methodically, problem solving can become second nature in tackling equation systems. Communication is also crucial, as it helps in verifying the solution's correctness and its adherence to the problem's context.