Problem 49 - Elementary Algebra

Question

A motel in a suburb of Chicago rents single rooms for \(\$ 62\) per day and double rooms for \(\$ 82\) per day. If a total of 55 rooms were rented for \(\$ 4210\), how many of each kind were rented?

Step-by-Step Solution

Answer

15 single rooms and 40 double rooms were rented.

Step 1: Define Variables

Let's define two variables to represent the number of single rooms and double rooms rented. Let \( x \) be the number of single rooms rented, and \( y \) be the number of double rooms rented.

Step 2: Create the System of Equations

We have two facts: the total number of rooms rented and the total revenue. We can create the following system of equations: 1. \( x + y = 55 \) (total rooms rented)2. \( 62x + 82y = 4210 \) (total revenue from renting the rooms)

Step 3: Solve the First Equation for a Variable

From the first equation \( x + y = 55 \), solve for \( x \): \( x = 55 - y \).

Step 4: Substitute into the Second Equation

Substitute \( x = 55 - y \) into the second equation: \[ 62(55 - y) + 82y = 4210 \]

Step 5: Simplify the Equation

Distribute the 62 in the equation: \[ 3410 - 62y + 82y = 4210 \]

Step 6: Combine Like Terms

Combine the \( y \) terms: \[ 3410 + 20y = 4210 \]

Step 7: Solve for \( y \)

Subtract 3410 from both sides to isolate \( y \): \[ 20y = 4210 - 3410 \] \[ 20y = 800 \] Now divide both sides by 20: \[ y = 40 \]

Step 8: Find \( x \) using \( y \) Value

Substitute \( y = 40 \) back into the equation \( x = 55 - y \): \( x = 55 - 40 = 15 \)

Step 9: Verify the Solution

Check that these values satisfy the original equations. For total rooms: \( 15 + 40 = 55 \). For total revenue: \( 62 \times 15 + 82 \times 40 = 930 + 3280 = 4210 \). Both conditions are met, confirming the solution is correct.

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving is a fundamental pillar of mathematics that allows us to find unknown values using known quantities. In this system of linear equations exercise, we define variables to represent unknowns: the number of single and double rooms rented. By establishing relationships between these variables—through algebraic expressions and equations—we can systematically determine their values.

The beauty of algebra lies in its ability to transform complex situations into manageable steps. This is achieved by:

• defining variables to represent unknowns,
• creating equations based on given data, and
• manipulating equations to isolate and solve for variables.

In our example, through nine detailed steps, we move from defining the problem to verifying our solution, ensuring an accurate and logical approach to problem-solving. This hands-on strategy can be applied to a wide variety of real-world challenges.

Equations with Two Variables

Working with equations involving two variables is a hallmark of systems of linear equations. These are mathematical tools used to represent scenarios where more than one unknown quantity exists.

In the provided exercise, we have two primary pieces of information: the total number of rooms rented and the total rental income. These clues give rise to two separate equations:

• The first equation (\( x + y = 55 \) ) reflects the total number of rooms rented.
• The second equation (\( 62x + 82y = 4210 \) ) accounts for the total revenue based on room types.

These equations enable us to solve for two variables: \( x \) (single rooms) and \( y \) (double rooms).

To solve this system, one common method is substitution. We express one variable in terms of the other and substitute this expression into the second equation. This reduces the equations to a single variable, which is much easier to handle, thereby simplifying our solution process. Once one variable is found, the method allows us to solve effortlessly for the other, ensuring that both pieces of the puzzle fit perfectly together.

Linear Equations in Real-World Applications

Linear equations are not just abstract mathematical concepts; they are powerful tools used to solve real-world problems. In daily life, many situations can be modeled using linear equations, helping us make informed decisions.

Consider the example of the motel; it demonstrates how linear equations allow business owners to analyze operations and revenues. With the given scenario, understanding how each type of room contributes to total revenue can provide insights into effective pricing and capacity management.

Linear equations simplify complex relationships into linear models, making it easier to predict outcomes, optimize processes, and make strategic decisions.

• They help determine costs versus benefits, analyze spending, and budget effectively.
• In business, they assist in evaluating sales and profits through demand-supply equations.
• In science, they model relationships between variables, like speed and distance.

Mastering linear equations equips individuals with the tools to translate everyday challenges into mathematical models, making analyzing and solving them a more straightforward task.