Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Self-Reported Spending The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was \(\$ 119.00\) in March \(2017 .\) High-income consumers spend \(\$ 88\) more than middle-/low- income consumers. (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle- Mow -income consumers. (b) Solve the system. (c) Interpret the solution.

Step-by-Step Solution

Answer

High-income consumers spend $163, while middle-/low-income consumers spend $75.

Step 1: Define the Variables

Let \( x \) be the average spending by high-income consumers and \( y \) be the average spending by middle-/low-income consumers.

Step 2: Establish the First Equation

The average spending for both groups combined is given as \( 119.00 \). Therefore, the equation representing this is: \[ \frac{x + y}{2} = 119.00 \] which simplifies to \( x + y = 238.00 \) since the average is calculated over two groups.

Step 3: Establish the Second Equation

High-income consumers spend \( 88 \) dollars more than middle-/low-income consumers. This can be represented as: \[ x = y + 88 \]

Step 4: Solve the System of Equations

We have the system of equations:1. \( x + y = 238.00 \)2. \( x = y + 88 \)Substitute \( x = y + 88 \) into the first equation:\( (y + 88) + y = 238.00 \)Combine like terms:\( 2y + 88 = 238.00 \)Subtract 88 from both sides:\( 2y = 150.00 \)Divide both sides by 2:\( y = 75.00 \)Substitute \( y = 75.00 \) back into \( x = y + 88 \):\( x = 75.00 + 88 \)\( x = 163.00 \)

Step 5: Interpret the Solution

The solution \( x = 163.00 \) and \( y = 75.00 \) tells us that high-income consumers self-reported spending \( 163.00 \) dollars on average yesterday, while middle-/low-income consumers self-reported spending \( 75.00 \) dollars on average.

Key Concepts

High-Income Consumers

High-income consumers generally have more disposable income and thus the ability to spend more freely compared to other groups. This can be influenced by higher salaries, investments, and a stronger financial position. In the context of our exercise, high-income consumers were found to spend an average of \(163.00\) dollars. This figure is significantly higher by \(88\) dollars compared to middle-/low-income consumers. For educators and students studying economics or consumer behavior, it’s crucial to understand how high-income status can impact spending patterns. These consumers tend to allocate more funds toward luxury goods, investment opportunities, and sometimes for lifestyle enhancements that can further improve their quality of life.
By analyzing their spending through linear equations, students can grasp how economic forces and individual financial capability influence aggregate spending patterns.

Middle-/Low-Income Consumers

Middle-/low-income consumers may have stricter budgets and prioritize essential goods and services over discretionary ones. Their spending behavior is often more conservative and focused on necessities due to financial constraints. In the exercise, it was observed that these consumers spent \(75.00\) dollars on average, which reflects their different economic situation compared to high-income consumers.
The interplay between different income groups is crucial in economic studies because these groups collectively drive market demand. Understanding their spending helps businesses to tailor products and services in alignment with their purchasing power. Moreover, for students, it's insightful to recognize the limitations that middle/low-income consumers might face, providing a real-world application of how economic equity and resource distribution play out in everyday scenarios.

Solving Linear Equations

Linear equations are powerful tools in mathematics used to represent and solve real-world problems involving relationships between different quantities. They are especially helpful in cases where you have known variables and need to find unknowns based on certain conditions.
In this exercise, we created a system of linear equations to determine the spending behaviors of two different consumer groups. The first equation \( x + y = 238.00 \) captures the total average spending, while \( x = y + 88 \) reflects the spending difference between the groups. Solving these equations with substitution uncovered the individual spending amounts, \( x = 163.00 \) for high-income consumers and \( y = 75.00 \) for middle-/low-income consumers.

• Step 1: Define variables to represent the quantities we are about to explore.
• Step 2: Formulate equations that encapsulate relationships.
• Step 3: Use methods like substitution or elimination to solve for unknowns.
• Step 4: Verify solutions to ensure they make sense within a real-world context.

By practicing linear equations, students enhance their ability to think analytically, an essential skill not just in math, but in daily decision-making and other academic disciplines as well.