Problem 87 - College Algebra with Modeling and Visualization

Question

Prices of Homes The median prices of a single-family home in the United States from 1990 to 2005 can be approximated by the formula \(P(x)=8667 x+90,000\) where \(x=0\) corresponds to 1990 and \(x=15\) to 2005 (Source: National Association of Realtors.) (a) Interpret the slope of the graph of \(P\). (b) Estimate the years when the median price range was from \(\$ 142,000\) to \(\$ 194,000\)

Step-by-Step Solution

Answer

Between 1996 and 2002, the median home price ranged from $142,000 to $194,000.

Step 1: Understanding the Slope

The function to approximate the median price of a home is given by \( P(x) = 8667x + 90,000 \). The slope of this linear function is 8667. In the context of this problem, the slope represents the rate of change of the price per year. Thus, the slope of 8667 means that the median price of a home increases by \( \$ 8667 \) each year.

Step 2: Set up equations for price estimation

We are tasked with finding the years when prices were between \( \\(142,000 \) and \( \\)194,000 \). We set up two equations: \( 8667x + 90,000 = 142,000 \) and \( 8667x + 90,000 = 194,000 \) to find the corresponding values for \( x \).

Step 3: Solve for the lower price limit

Solve the equation \( 8667x + 90,000 = 142,000 \) to find the initial year when the price reached \( \\(142,000 \):\[8667x + 90,000 = 142,000 \8667x = 142,000 - 90,000 \8667x = 52,000 \x = \frac{52,000}{8667} \approx 6\]So the median price first reached \( \\)142,000 \) around \( x = 6 \), which corresponds to the year 1996.

Step 4: Solve for the upper price limit

Now solve the equation \( 8667x + 90,000 = 194,000 \) for \( x \):\[8667x + 90,000 = 194,000 \8667x = 194,000 - 90,000 \8667x = 104,000 \x = \frac{104,000}{8667} \approx 12\]So the median price reached \( \$194,000 \) around \( x = 12 \), which corresponds to the year 2002.

Step 5: Determine the range of years

The median price of a home was between \( \\(142,000 \) and \( \\)194,000 \) from approximately 1996 (when \( x = 6 \)) to 2002 (when \( x = 12 \)).

Key Concepts

Slope Interpretation

In any linear function, the slope is a crucial element that tells us how one variable changes in relation to another. For the function \( P(x) = 8667x + 90,000 \), which is used to approximate the median price of a single-family home in the US from 1990 to 2005, the slope is 8667. So, what does this number really mean?
Simply put, it's a way to quantify the increase in home prices year after year. Here, the slope of 8667 indicates that the median price increases by \( \$8667 \) every year. You can think of the slope as the speed at which the home prices climb over time.
Understanding slopes is a fundamental skill, as it helps you not only in real estate calculations but in analyzing trends in various personal finance, science, and economic contexts.

Price Estimation

Price estimation involves predicting or identifying price points based on a given function. In this exercise, we're estimating during which years the median home price was between \( \\(142,000 \) and \( \\)194,000 \). This requires setting up equations to determine the years when prices hit these marks.
For the lower price limit of \( \\(142,000 \):

• You start with the equation \( 8667x + 90,000 = 142,000 \).
• Solve for \( x \) by isolating it: \( 8667x = 142,000 - 90,000 \).
• This simplifies to \( 8667x = 52,000 \), resulting in \( x \approx 6 \), which corresponds to the year 1996.

For the upper price limit of \( \\)194,000 \):

• Set up the equation \( 8667x + 90,000 = 194,000 \).
• Solve for \( x \) to find \( 8667x = 194,000 - 90,000 \).
• This leads to \( 8667x = 104,000 \), giving \( x \approx 12 \), approximately the year 2002.

So, we estimate that the median home price was between \( \\(142,000 \) and \( \\)194,000 \) from around 1996 to 2002.

Solving Equations

Solving linear equations is integral when forecasting values in mathematical models. These equations typically take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable you want to find. Here's how you solve them:
1. **Isolate the variable**: Start by moving all terms involving \( x \) to one side of the equation, and numeric constants to the other. This is done by subtracting or adding the numeric constant on the side with \( x \).
2. **Simplify**: Divide or multiply as needed to solve for \( x \). This involves eliminating any coefficient from \( x \) by performing the inverse operation. For example, if the equation is \( ax = d \), divide both sides by \( a \) to find \( x \).
In our problem:
For \( 8667x + 90,000 = 142,000 \) or \( 194,000 \):

• Subtract 90,000 from both sides to get \( 8667x \) alone.
• Solve by dividing by 8667 to determine \( x \).

This process helps translate a static equation into useful insights and provides a clear timeline for price changes within a defined parameter set.