In 2010, the average price of a new home sold in the United...

In $2010,$ the average price of a new home sold in the United States was $\$ 272,900 .$ In $2005,$ the average price of a new home in the United States was $\$ 297,000 .$ Let $y$ be the average price of a new home in the year $x,$ where $x=0$ represents the year 2005 (Source: Based on data from U.S.census) a. Write a linear equation that models the average price of a new home in terms of the year $x$. [ Hint: The line must pass through the points $(0,297,000) \text { and }(5,272,900) .]$ b. Use this equation to predict the average price of a new home in 2013 .

In $2010,$ the average price of a new home sold in the United States was $\$ 272,900 .$ In $2005,$ the average price of a new home in the United States was $\$ 297,000 .$ Let $y$ be the average price of a new home in the year $x,$ where $x=0$ represents the year 2005 (Source: Based on data from U.S.census)
a. Write a linear equation that models the average price of a new home in terms of the year $x$. [ Hint: The line must pass through the points $(0,297,000) \text { and }(5,272,900) .]$
b. Use this equation to predict the average price of a new home in 2013 .

Key Concepts

Linear Functions

A linear function represents a constant rate of change between two variables and is expressed in the form y = mx + b. It is used to model relationships where the change in one variable corresponds to a consistent change in another. This concept is foundational in algebra and helps simplify complex real-world situations into manageable mathematical models.

Slope Calculation

The slope of a linear function quantifies the rate at which y changes with respect to x, often interpreted as rise over run. Calculated as the difference in y-values divided by the difference in x-values between two points, the slope is key to determining the direction and steepness of the line, making it an essential step in formulating the linear equation.

Intercept Interpretation

The y-intercept of a linear function is the value of y when x equals 0, representing the starting point of the modeled scenario. Understanding the y-intercept is crucial because it provides a reference point from which the subsequent changes (dictated by the slope) are measured.

Extrapolation

Extrapolation involves using a mathematical model, such as a linear function, to predict the value of a variable outside the range of the original data points. It is a practical application of the model, though it comes with the assumption that the established linear trend continues beyond the observed data.

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