Problem 70 - Algebra for College Students

Question

Solve each problem by using a system of equations. An apartment complex rents one-bedroom apartments for \$325 per month and two- bedroom apartments for \(\$ 375\) per month. One month the number of onebedroom apartments rented was twice the number of two-bedroom apartments. If the total income for that month was \(\$ 12,300\), how many apartments of each kind were rented?

Step-by-Step Solution

Answer

24 one-bedroom and 12 two-bedroom apartments were rented.

Step 1: Define Variables

Let \( x \) represent the number of one-bedroom apartments rented and \( y \) represent the number of two-bedroom apartments rented.

Step 2: Create Equations

The problem provides two conditions that allow us to create two equations. 1. One-bedroom apartments are twice as many as the two-bedroom apartments: \[ x = 2y \]2. The total rent income is \$12,300. This translates to the equation: \[ 325x + 375y = 12300 \]

Step 3: Substitute Variables

Using the first equation \( x = 2y \), substitute \( x \) in the second equation.\[ 325(2y) + 375y = 12300 \]

Step 4: Simplify and Solve for y

Simplify the equation from Step 3:\[ 650y + 375y = 12300 \]\[ 1025y = 12300 \]Now, solve for \( y \):\[ y = \frac{12300}{1025} = 12 \]

Step 5: Solve for x

Use the value of \( y \) found in Step 4 to find \( x \) using the first equation:\[ x = 2y = 2 \times 12 = 24 \]

Step 6: Verify Solution

Verify the solution by substituting \( x = 24 \) and \( y = 12 \) back into the income equation:\[ 325(24) + 375(12) = 7800 + 4500 = 12300 \]Since both the equations are satisfied, the solution is correct.

Key Concepts

Linear Equations

Linear equations play a crucial role in algebra and help us find unknown variables based on given relationships. In the exercise, the two bedroom types and their costs are modeled using linear equations. These equations are represented in the form:

• Equation one: One-bedroom apartments are double the two-bedroom apartments: \[ x = 2y \]
• Equation two: The total income from rented apartments is \( 12300 \) dollars: \[ 325x + 375y = 12300 \]

Each of these equations represents a straight line when graphed, as they relate the quantities of each apartment type to their respective monthly rentals.

Solution Verification

Once you have solved the equations in a system, it is essential to verify the solution. Verification ensures that the solution satisfies the original conditions of the problem. In our example, we do this by plugging the values of \( x = 24 \) and \( y = 12 \) back into the income equation:

• Calculate the total income using the found values: \[ (325 \times 24) + (375 \times 12) = 7800 + 4500 = 12300 \]
• Ensure the total matches the original income constraint of \( 12300 \) dollars.

Verification is a critical step as it confirms the accuracy and correctness of the obtained solution after the calculation process.

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves replacing one variable with an expression derived from another equation. Here’s how it works in this exercise:

• Firstly, express one variable in terms of the other. From \( x = 2y \), we know \( x \) in terms of \( y \).
• Next, substitute \( x = 2y \) into the second equation \[ 325(2y) + 375y = 12300 \]. This step reduces two-variable equations into a single-variable equation.
• The substitution method simplifies the problem, making it easier to solve for the remaining variable.

Variable Definition

Defining variables is the starting point of solving any system of equations. By assigning symbols to unknown quantities, we can represent real-world problems mathematically. In the problem, the variables \( x \) and \( y \) are used:

• \( x \) represents the number of one-bedroom apartments rented.
• \( y \) is the number of two-bedroom apartments rented.

Proper variable definition is crucial as it sets the foundation for developing equations that mirror the problem scenario. Furthermore, it assists in maintaining clarity and consistency throughout the problem-solving process.