Question

You are given the 2005 value of a product and the rate at which the value is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year. (Let \(t=5\) represent 2005.) 2005 Value \(\quad\) Rate $$\$ 20,400 \quad \$ 2000$$ decrease per year

Short answer

The linear equation that gives the dollar value of the product in terms of the year is: Value of product = -\$2000t + \$20,400. This indicates that each year, the value of the product decreases by \$2000.

Step-by-step solution

Identify the initial value and the rate of change

The initial value in 2005 is given as \$20,400. This is the y-intercept of the linear equation (the value of the y when t=0). The rate of change is a decrease of \$2000 per year. Because this is a decrease, the rate of change is -\$2000.

Construct the linear equation

A linear equation can be written in the form y = mx + b, where m is the slope (rate of change), and b is the y-intercept (initial value). The rate of change (m) is -\$2000, and the initial value (b) is \$20,400. Therefore, the linear equation is y = (-\$2000)t + \$20,400.

Replace y with the value of the product

In this real-world context, y is the value of the product. So the equation becomes: Value of product = -\$2000t + \$20,400.

Key concepts

Rate of Change

The concept of 'rate of change' is fundamental to understanding how quantities vary with respect to one another. In the context of linear equations, the rate of change represents how much one variable changes for a one-unit increase in another variable. In the given exercise, we are told the value of a product decreases by $2000 each year, making our rate of change a negative value, specifically \( -\$2000 \). This attribute of the linear equation is crucial because it defines the steepness or the slant of the line when graphed on a coordinate plane and encapsulates the speed at which the value of the product is expected to decline over time. Understanding the concept of the rate of change allows us to predict future values, not only in this particular financial scenario but in diverse applications such as physics for velocity or economics for cost analysis.

When visualizing the scenario, if we plot the year on the x-axis and the value of the product on the y-axis, the line will slope downward from left to right, indicating a decrease in value over time due to the negative rate of change.

Y-intercept

The 'y-intercept' is a specific point where the graph of an equation crosses or touches the y-axis of a coordinate plane. It occurs where the independent variable (in this case, time, represented as \( t \)) equals zero. This point provides us with a snapshot of the initial state of the variable we're examining before any change occurs. For the provided exercise, the y-intercept represents the initial value of the product in 2005, which is \( \$20,400 \).

In practical terms, the y-intercept gives us a starting point for our linear graph. For example, before any time has passed, we will know exactly the value of our product just by looking at this point on the graph. In various fields, the y-intercept carries significant meaning: in finance, it could represent the initial investment; in physics, the starting position of a moving object; and in population studies, the size of a population at the beginning of the study.

Slope

The 'slope' of a linear equation is the number that expresses the tilt or incline of the line on a graph. It's numerically equal to the rate of change and is represented by the letter \( m \) in the slope-intercept form of a linear equation, which is \( y = mx + b \). The slope describes how many units the y variable (the product's value) moves up or down for each unit of movement along the x variable (time).

In our exercise, the slope is the negative rate at which the product's value decreases per year, \( -\$2000 \). A positive slope would imply an upward trend in value over time, whereas our negative slope indicates a decrease. It's the reason why in the graph of this linear equation, the line will descend as we move from left to right. The steeper the slope, the faster the value changes. Grasping the concept of slope is essential for predicting future outcomes, comparing rates of change, and understanding the overall behavior of linear functions.

Algebraic Representation

When we discuss 'algebraic representation', we're referring to the expression of relationships between variables using algebraic equations. These representations help us understand and manipulate mathematical models to solve real-world problems. In our linear equation example, the algebraic representation is \( y = -\$2000t + \$20,400 \), where \( y \) represents the dollar value of the product and \( t \) denotes the number of years after 2005.

The algebraic representation is particularly powerful because it allows us to compute the value of the product for any given year simply by substituting the corresponding value of \( t \) into the equation. It is concise, yet holds all the information about the rate at which the value decreases (slope) and the initial value in 2005 (y-intercept), making it a vital tool for analysis and prediction. An understanding of algebraic representation is not only fundamental in mathematics but it is also extensively used in fields such as engineering, science, economics, and data analysis.