Problem 50 - Algebra 1

Question

The science club is selling magazine subscriptions at \(\$ 15\) each. The club wants to raise \(\$ 315\) for science equipment. Assign labels and write an algebraic model based on your verbal model.

Step-by-Step Solution

Answer

The algebraic model based on the verbal model is \( 15x = 315 \), where \( x \) represents the number of magazine subscriptions sold.

Step 1: Assign Variables

Assign a variable to describe the number of magazine subscriptions. Let \( x \) be the number of subscriptions.

Step 2: Define Relationship

The total amount raised is product of the price per subscription, which is $15, and the number of subscriptions sold. Hence we can write this relation as \( 15x \).

Step 3: Formulate Algebraic Model

The club plans to raise $315. This is equal to the total amount raised. Hence, the algebraic model is \( 15x = 315 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent a line when graphed on a coordinate plane. They take the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. These equations are fundamental in algebra as they describe a straight line's behavior. Understanding how to construct and solve them is crucial in various fields, from science to economics.

• In our exercise, the linear equation is \( 15x = 315 \), which is derived from multiplying the price per subscription by the number of subscriptions.
• The constant \( 15 \) represents the cost of each subscription, while \( 315 \) signifies the fundraising target.

Solving a linear equation frequently involves isolating \( x \) on one side to find the number of solutions that meet the condition set by the equation.

Problem-Solving Strategies

Applying effective problem-solving strategies can simplify complex mathematical problems. Here are some steps you can follow:

• **Identify the Problem**: Clearly understand what is being asked. In this case, we need to calculate the number of magazine subscriptions required to achieve the fundraising goal of $315.
• **Devise a Plan**: Break down the problem into smaller parts. We decided on assigning variables first, which paved the way for forming an equation.
• **Execute the Plan**: Implement the steps you've laid out. Use the variable \( x \) for the number of subscriptions, and create your equation: \( 15x = 315 \).
• **Evaluate the Solution**: Once you solve the equation, ensure your solution makes sense. Solve \( x \) and verify: \( x = \frac{315}{15} = 21 \). Thus, 21 magazine subscriptions need to be sold.

Adopting these strategies will improve your problem-solving capabilities and help develop robust critical thinking skills.

Variable Assignment

Variable assignment is a fundamental concept in algebra that involves setting a letter or symbol to represent a number. This approach makes it easier to form equations and solve problems systematically.
In the given problem:

• We assigned \( x \) to represent the number of magazine subscriptions sold.
• Choosing 'x' or another letter is common in algebra as it serves as a placeholder for the unknown quantity.

Variable assignment is not only helpful in making abstract concepts tangible but is also essential in building the algebraic model. By correctly assigning and using variables, complex word problems become straightforward mathematical equations, enabling us to find solutions efficiently.