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Chapter 2: Problem 71

Assume a linear relationship holds. In 1975 , an average house in San Jose cost $$\$ 45,000$$ and the same house in 1995 costs $$\$ 195,000$$. Write an equation that will give the price of a house in any year, and use this equation to predict the price of a similar house in the year 2010 .

Short Answer

Expert verified
The equation of the line that gives the house price depending on the year is \(y = \$7500x - \$14425000\). Based on this equation, the predicted price for a similar house in 2010 is \$3765000.

Step by step solution

01

Identify Knowns

The coordinate pairs are as follows: (1975, \$45000) and (1995, \$195000). It is important to note that the years represent the x-values, while the prices represent the y-values.
02

Determine the Slope of the Line

The slope of a line is calculated as the change in the y-values divided by the change in the x-values. Formally, it can be written as \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). Substituting from the known points gives \(m = \frac{195000 - 45000}{1995 - 1975} = \$7500\). Therefore, the slope of our line is \$7500.
03

Determine the Equation of the Line

The equation of a line is given by \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Given the slope calculated in Step 2, one of our known points and rearranging the equation to solve for \(b\), we get, \(b = y - mx\). Substituting from (1975 , \$45000), \(b = \$45000 - \$7500 * 1975 = -\$14425000\). Consequently, our equation becomes \(y = \$7500x - \$14425000\).
04

Predict the House Price in 2010

Now that we have the equation, we can use it to predict the price of the house in 2010. Substituting \(x = 2010\) into the equation gives: \(y = \$7500 * 2010 - \$14425000 = \$3765000\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form is essential for writing linear equations. This form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. The slope (\

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Most popular questions from this chapter

Assume a linear relationship holds. An average math text book cost $$\$ 25$$ in $$1980,$$ and $$\$ 60$$ in $$1995 .$$ Write an equation that will give the price of a math book in any given year, and use this equation to predict the price of the book in 2010 .

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