Problem 51 - Algebra 2

Question

Nathan has 50 baseball cards in his collection from the \(1950^{\prime} \mathrm{s}\) and \(1960^{\prime} \mathrm{s}\) . His goal is to buy 2 more cards each month. Write an equation that represents how many cards Nathan will have in his collection in \(x\) months if he meets his goal.

Step-by-Step Solution

Answer

Nathan will have \( 50 + 2x \) baseball cards after \( x \) months.

Step 1: Understanding the Problem

Nathan starts with 50 baseball cards. His goal is to buy 2 more cards each month. We need to find an equation to represent the total number of cards after \( x \) months.

Step 2: Identifying Variables

Let \( x \) represent the number of months Nathan is collecting cards. We are asked to determine how many total cards he has after these months.

Step 3: Setting up the Equation

Since Nathan starts with 50 cards and buys 2 new cards each month, we combine these two pieces of information to form the equation. If he buys 2 cards per month, after \( x \) months, he will have bought \( 2x \) cards in total.

Step 4: Writing the Equation

Add the number of cards he started with to the number of cards he purchases over \( x \) months: \( 50 + 2x \). This represents the total number of baseball cards Nathan will have.

Key Concepts

Variables in Algebra

In algebra, variables are symbols or letters that represent unknown values in mathematical expressions and equations. They act as placeholders, enabling us to create general formulas.Variables are crucial as they allow patterns and relationships to be expressed in simplistic forms.

In the original exercise, we see an example where Nathan's collection of baseball cards depends on time. Here, the variable is denoted as \(x\). It stands for the number of months Nathan is collecting cards. By using \(x\), we can form an equation that models how his card count changes over time. Variables like \(x\) allow us to consider varying scenarios by plugging in different values without changing the entire equation.

• Variable serve as a versatile tool to tackle different mathematical problems.
• They provide a compact and general way to express mathematical relationships.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Equations are like statements of equality. They tell us that two mathematical expressions represent the same value.

In our exercise, we created the equation \(50 + 2x\). It models Nathan's goal of collecting baseball cards over several months. Here, solving it doesn't require isolating \(x\) like some equation types because it’s already expressed for various \(x\) values.

• Simple equations typically involve basic arithmetic operations like addition, subtraction, or multiplication.
• More complex equations could require techniques such as cross-multiplying or factoring.

Solving this equation involves choosing a specific \(x\), inserting it into the equation, and calculating. For example, if \(x=3\), Nathan would have \(50 + 2(3) = 56\) cards.

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations with mathematical formulas or equations. This allows us to analyze, make predictions, and understand relationships between different factors.

In Nathan’s case, we took his collection strategy and created the equation \(50 + 2x\). This models how his collection changes over time. It helps us answer questions like, "How many cards will Nathan have after 10 months?" by inputting the value \(x = 10\) into the equation.

• Modeling helps us simplify and solve practical problems.
• It often starts with identifying known quantities and establishing relationships between them.

The equation is a simple representation but shows how mathematical modeling aids in understanding growth over time, reflecting its usefulness in planning and decision-making.By understanding what each part of the equation represents, we can adapt it to similar scenarios, whether it be savings, distance, or growth over time.