Problem 66 - Algebra A Combined Function

Question

Solve. See Example 7. In 2007 , IBM had 121,000 employees in the United States. By \(2009,\) this number had shrunk to 105,000 . Let \(y\) be the number of IBM U.S. employees in year \(x,\) where \(x=0\) represents 2007. (Source: Computerworld) a. Write a linear equation that models the number of IBM employees in the year \(x\). b. Use this equation to estimate the number of IBM U.S. employees for the year 2011 .

Step-by-Step Solution

Answer

89,000 IBM U.S. employees are estimated for 2011.

Step 1: Understand the Problem

We have two data points: In 2007 (\(x = 0\)), the number of employees was 121,000, and in 2009 (\(x = 2\)), it was 105,000. We need to find a linear equation and use it to estimate the number of employees in 2011 (\(x = 4\)).

Step 2: Find the Slope

The slope (\(m\)) of the linear equation is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points (0, 121,000) and (2, 105,000), we calculate: \[m = \frac{105,000 - 121,000}{2 - 0} = \frac{-16,000}{2} = -8,000.\]

Step 3: Write the Linear Equation

The linear equation in slope-intercept form is \(y = mx + b\). We have found \(m = -8,000\). To find \(b\), we use one of the points, e.g., (0, 121,000):\[121,000 = -8,000 \times 0 + b\]\[b = 121,000.\]Thus, the equation is \(y = -8,000x + 121,000.\)

Step 4: Estimate Employees in 2011

Since 2011 corresponds to \(x = 4\), substitute \(x = 4\) into the equation:\[y = -8,000 \times 4 + 121,000\]\[y = -32,000 + 121,000\]\[y = 89,000.\] The estimated number of employees in 2011 is 89,000.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to describe a straight line on a graph using an easy formula. This formula is given by \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.

This form is particularly useful because it instantly tells us two essential things:

• The slope \( m \), which shows how steep the line is.
• The y-intercept \( b \), indicating the starting point of the line on the y-axis.

When dealing with real-world problems, this formula allows us to quickly understand and predict trends. We can use it to find values of \( y \) when we know \( x \), making it ideal for problems involving change over time, like the decrease in the number of employees over the years.

Understanding the slope-intercept form is key to tackling many algebraic problems efficiently.

Linear Modeling

Linear modeling is a process of creating equations or formulas to describe real-world phenomena, allowing us to make predictions or understand trends. In the context of this exercise, a linear model was used to represent the change in IBM’s employee numbers over time.

Creating a linear model involves:

• Identifying two key data points that define the change.
• Calculating the slope to see how one quantity changes in relation to another.
• Constructing the model using the slope-intercept form, \( y = mx + b \).

The advantages of linear modeling are significant. It's simple yet powerful, making complex data easier to digest. For businesses and analysts, it serves as a tool to forecast future scenarios based on historical data, such as predicting the workforce in future years.

In real-life applications, linear models can be used in economics, population studies, and any field where trends over time are analyzed. For students, mastering linear modeling is critical for success in further mathematical studies and practical problem-solving.

Slope Calculation

Slope calculation is a fundamental concept in algebra, representing the rate of change between two variables. The formula for finding the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This calculation is crucial in understanding how two data points connect on a graph.

The slope is often described as "rise over run," which means:

• "Rise" is the change in the y-values (vertically).
• "Run" is the change in the x-values (horizontally).

In the exercise provided, the slope tells us how the number of IBM employees decreased each year. A negative slope indicates a decline. Here, the slope was \(-8,000\), meaning the number of employees decreased by 8,000 each year.

Understanding how to calculate and interpret the slope is vital. It helps identify patterns and predict future values based on past trends, making it an indispensable tool in both academic and real-world analyses.