Problem 7 - Algebra 2

Question

SHOPPING For Exercises \(7-9,\) use the following information. Gwen wants to buy some used CDs that cost \(\$ 10\) each and some used DVDs that cost \(\$ 13\) each. She has \(\$ 40\) to spend. Write an inequality to represent the situation, where \(c\) is the number of CDs she buys and \(d\) is the number of DVDs.

Step-by-Step Solution

Answer

The inequality is \(10c + 13d \leq 40\).

Step 1: Understand the Problem

Gwen wants to purchase CDs and DVDs within a budget of \(40. The cost of each CD is \)10, and each DVD is $13. We need to express her budget limitations as an inequality in terms of the number of CDs \(c\) and DVDs \(d\) she can purchase.

Step 2: Create an Equation for the Total Cost

The total cost for CDs is given by \(10c\) and for DVDs by \(13d\). Hence, the expression for the total amount spent is \(10c + 13d\).

Step 3: Formulate the Inequality

Since Gwen has a budget of $40, the total cost for purchasing CDs and DVDs should not exceed this amount. Therefore, the inequality is \(10c + 13d \leq 40\).

Key Concepts

Understanding Budget Constraints

Budget constraints are a critical part of financial decision-making. Imagine you have a limited amount of money to spend, just like Gwen in our problem. She has \(\\(40\) to use for buying CDs and DVDs. The idea here is to make sure that the total amount she spends does not go beyond her budget.
Think of a budget constraint as a boundary you should not cross. It helps you determine the combination of items you can afford. For Gwen, each CD costs \(\\)10\) and each DVD costs \(\\(13\). Her budget constraint limits her choices to a sum where the combined cost of CDs and DVDs does not exceed \(\\)40\).
When working with budget constraints, you always want to keep in mind the available amount and prioritize spending according to needs and preferences.

Working with a System of Inequalities

A system of inequalities involves multiple inequalities that need to be satisfied simultaneously. In Gwen’s case, this system consists of combinations of CDs and DVDs she can purchase without exceeding her budget.
Here's how Gwen's situation works:

• The inequality \(10c + 13d \leq 40\) represents all the possible combinations where the total cost of CDs \((10c)\) and DVDs \((13d)\) does not surpass \(\$40\).
• An additional constraint would be that both \(c\) and \(d\) should be non-negative integers since you can't purchase a negative number of items."

Creating a system helps visualize and analyze the various purchasing scenarios available to Gwen. It simplifies decision-making by narrowing down feasible options under given constraints.

Exploring Linear Equations

Linear equations form the backbone of many mathematical models like inequalities. They involve variables that show up in a straight line when graphed on a coordinate plane. In Gwen's scenario, the terms \(10c\) and \(13d\) are part of a linear relationship reflecting the costs of CDs and DVDs.
With the equation \(10c + 13d = \text{Total Cost}\), you can determine any specific expense outcome from Gwen’s budget plan. If Gwen decided to spend her entire \(\$40\), this translates to solving the linear equation \(10c + 13d = 40\).
Linear equations are the perfect tool for mapping out scenarios when working within budget constraints, highlighting the maximum and minimum possibilities according to various spending choices.

Strategies for Problem Solving

Problem-solving is a valuable skill when confronted with real-life scenarios like Gwen's shopping budget. Solving such problems involves breaking down the situation into manageable parts:

• Identify the terms: Clarify what each variable represents. Here, \(c\) represents CDs and \(d\) stands for DVDs.
• Establish the relationships: Understand how costs and budget relate through the equation \(10c + 13d \leq 40\).
• Test possibilities: Plug in different combinations for \(c\) and \(d\) to find valid options that keep the cost within the budget.

These steps help in creating a structured approach to tackling the problem, ensuring no potential outcomes are overlooked. Solid problem-solving strategies are crucial in decision-making, especially when dealing with financial choices.