Problem 38 - College Algebra with Modeling and Visualization

Question

Home Prices Prices of homes can depend on several factors such as size and age. The table shows the selling prices for three homes. In this table, price \(P\) is given in thousands of dollars, age \(A\) in years, and home size \(S\) in thousands of square feet. These data may be modeled by \(P=a+b A+c S\) $$ \begin{array}{ccc} \hline \text { Price (P) } & \text { Age (A) } & \text { Size (S) } \\ \hline 190 & 20 & 2 \\ 320 & 5 & 3 \\ 50 & 40 & 1 \end{array} $$ (a) Write a system of linear equations whose solution gives \(a, b,\) and \(c\) (b) Solve this system of linear equations. (c) Predict the price of a home that is 10 years old and has 2500 square feet.

Step-by-Step Solution

Answer

The predicted price for a 10-year-old, 2500 sq ft home is $260,000.

Step 1: Define the System of Equations

The equation given is a linear model: \(P = a + bA + cS\). For each home, substitute the values of \(P\), \(A\), and \(S\) into the equation.- Home 1: \(190 = a + 20b + 2c\)- Home 2: \(320 = a + 5b + 3c\)- Home 3: \(50 = a + 40b + c\)This gives us a system of three linear equations in terms of \(a\), \(b\), and \(c\).

Step 2: System of Linear Equations

We have the following system of equations:1. \(a + 20b + 2c = 190\)2. \(a + 5b + 3c = 320\)3. \(a + 40b + c = 50\)

Step 3: Solve for 'a' using Elimination Method

To simplify the system, subtract the second equation from the first:\((a + 20b + 2c) - (a + 5b + 3c) = 190 - 320\)\(15b - c = -130\) — (Equation 4)Similarly, subtract the third equation from the first:\((a + 20b + 2c) - (a + 40b + c) = 190 - 50\)\(-20b + c = 140\) — (Equation 5)

Step 4: Solve for 'b' and 'c' using Linear Combination

Add Equation 4 and Equation 5 to eliminate \(c\):\(15b - c + (-20b + c) = -130 + 140\)\(-5b = 10\)\(b = -2\)Substitute \(b = -2\) back into Equation 4:\(15(-2) - c = -130\)\(-30 - c = -130\)\(c = 100\)

Step 5: Solve for 'a'

Now that we have \(b = -2\) and \(c = 100\), substitute back into the first original equation:\(a + 20(-2) + 2(100) = 190\)\(a - 40 + 200 = 190\)\(a + 160 = 190\)\(a = 30\)

Step 6: Predict the Price of a 10-year old 2500 square foot Home

Using \(a = 30\), \(b = -2\), and \(c = 100\), plug these values into the model for a 10-year-old home with a size of 2500 square feet (2.5 in thousands):\(P = 30 + (-2)(10) + 100(2.5)\)Calculate:\(P = 30 - 20 + 250\)\(P = 260\)Thus, the predicted price is $260,000.

Key Concepts

Linear Models

A linear model is a mathematical equation that represents a straight-line relationship between variables. In the context of home pricing, this model helps us understand how factors such as the age and size of a home impact its price. Linear models use an equation of the form:

• \( P = a + bA + cS \)

Here, \( P \) is the dependent variable representing the home's price. The variables \( A \) (age) and \( S \) (size) are the independent variables that influence \( P \). The coefficients \( a, b, \) and \( c \) reflect the underlying relationship between these factors and the price.
For example, the coefficient \( a \) is the baseline price that doesn't depend on age or size. Meanwhile, \( b \) shows how much the price decreases or increases with each additional year in the home's age. Similarly, \( c \) tells us how the price changes per unit increase in home size. By adjusting these coefficients appropriately, we can tailor the linear model to fit real-world data and make predictions.

Predictive Modeling

Predictive modeling involves using statistical techniques like linear models to forecast future outcomes based on historical data. The primary goal of predictive modeling is to understand current patterns and make informed predictions about future events.
In our exercise, we've used a linear model to derive a predictive model that estimates the price of homes based on age and size. This model, once calibrated with the data of existing homes, allows us to answer the question of how much a new home might cost.

• We start by setting up a system of equations based on the linear model and solving for the coefficients \( a, b, \) and \( c \).
• After determining these coefficients, we can input new values for age and size to predict home prices.

Predictive models are valuable tools in economics and real estate because they can provide estimates that help buyers, sellers, and investors make financial decisions.

Home Price Analysis

Home price analysis involves evaluating the various factors that affect the price of a house. This analysis helps us establish patterns and trends that are crucial for making informed real estate decisions. In our home price exercise, the main factors considered are the age and size of the home.
Age plays a significant role because it can imply required renovations or the decreased desirability of older homes, despite historical value. This is represented in our linear equation as a variable that can either reduce or increase the price.
Size is directly correlated with price since larger homes offer more living space, generally increasing the market value. Our analysis shows that while these factors definitely influence prices, the model also helps quantify their exact impact using coefficients.

• After analyzing the data and understanding the model's parameters, we can better understand how these factors individually and collectively impact home prices.
• This analysis can guide potential homeowners in figuring out the fair market value of a property.

By leveraging these insights, buyers and sellers can have a more objective discussion about pricing, offering a clearer picture of the real estate landscape.