Problem 84 - Algebra A Combined Function

Question

The average price of an acre of U.S. farmland was \(\$ 1210\) in 2002 . In 2008 , the price of an acre rose to \$2350. (Source: National Agricultural Statistics Services) a. Write two ordered pairs of the form (year, price of an acre). b. Find the slope of the line through the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

Step-by-Step Solution

Answer

(2002, 1210), (2008, 2350); slope = 190; price increased by $190 per year.

Step 1: Identify Ordered Pairs

First, identify the ordered pairs in the form (year, price of an acre) using the given data. For the year 2002, when the price was $1210, the ordered pair is (2002, 1210). For the year 2008, with a price of $2350, the ordered pair is (2008, 2350). Therefore, the two ordered pairs are (2002, 1210) and (2008, 2350).

Step 2: Calculate the Slope

To find the slope of the line through the two points, use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1) = (2002, 1210)\) and \((x_2, y_2) = (2008, 2350)\).Substituting the values, we get\[ m = \frac{2350 - 1210}{2008 - 2002} = \frac{1140}{6} = 190 \].Thus, the slope is 190.

Step 3: Interpret the Slope

The slope of the line, 190, represents the rate of change in the price of an acre of farmland per year from 2002 to 2008. Specifically, the price increased by $190 each year during this period.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in math, particularly when dealing with graphs and coordinates. An ordered pair is written in the form

• \((x, y)\), where each element is called a coordinate.

The first number, often referred to as the \(x\)-coordinate, represents a position along the horizontal axis of a graph while the second number, or \(y\)-coordinate, corresponds to the vertical axis. In the context of the exercise, ordered pairs are used to represent the year and the price of an acre of farmland. Using the given data, we have the ordered pairs of

• (2002, 1210) and
• (2008, 2350).

These pairs are helpful because they provide a precise way to pinpoint specific data points—in this case, how much the price of farmland was during a particular year.

Rate of Change Explained

The term "rate of change" is often used interchangeably with "slope" in linear equations. Rate of change is a measure of how one quantity changes in relation to another. To grasp this idea better, imagine the slope as a hill or a road's incline.The calculations used in the example show that the slope of the line through the points

• (2002, 1210)
• and (2008, 2350)

is 190. By using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\] where \(x\) represents the year and \(y\) stands for the price, we determine the rate at which prices increased each year. When calculated, the 190 indicates that the price increased by $190 per year between 2002 and 2008. This concept helps us understand trends over time, like how farmland prices increased each year in this case study.

Introduction to Linear Equations

Linear equations can be thought of as a straight line when graphed. They generally take the form of \[y = mx + b\]where:\[\text{- \(m\) is the slope} \]and \[\text{- \(b\) is the \(y\)-intercept}\]which is where the line crosses the \(y\) axis. Linear equations are significant in various real-world scenarios, especially where you have a constant rate of change, like the consistent increase in farmland prices. By determining the slope, you now have part of a linear equation. For this exercise, if you also knew the starting value when the \(x = 0\), you could write the full linear equation representing farmland prices over time. This type of math makes analyzing data more systematic and offers insights into future predictions.