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 $$\$ 154,000 \quad \$ 12,500$$ increase per year

Short answer

The linear equation is \(v = 154000 + 12500(t-5)\). This equation gives the dollar value of the product in terms of the year, where \(t\) represents the number of years after 2005.

Step-by-step solution

Establish the Variables

Let's start by defining 't' as the number of years after 2005 and 'v' as the value of the product. In 2005, t equals 5 and v is $154,000.

Understand the Rate of Increase

The problem indicates that the value of the product increases by $12,500 each year. This is a key piece of information. It means that after 1 year from 2005, the product's value will be the initial value plus $12,500. This pattern will hold for each subsequent year.

Formulate the Linear Equation

Using these data, we can establish a linear equation to represent the value of the product per year. The product value in any year equals the 2005 value ($154,000) plus the rate of increase ($12,500) multiplied by the number of years after 2005. Therefore, the linear equation will be: \(v = 154000 + 12500(t-5)\)

Key concepts

Rate of Change

In the context of linear equations, the rate of change represents how much a quantity increases or decreases over a unit of time. Think of it as the 'speed' at which a value changes. Here, the rate of change is the annual increase in the product's value.
For example, if a product's value grows by $12,500 each year, this means that each year, starting from a baseline (in this case, the year 2005), the value increases by $12,500. The rate of change provides a consistent pattern or trend of change, allowing us to predict future values.
This concept is essential for creating linear equations because it helps determine the slope of the equation. If we imagine a graph, the rate of change defines the steepness or incline of the line, with a positive rate indicating an upward trend.
In our problem, it means that if you know the value of the product in one year, you can easily find its value in another year by adding the rate of change multiplied by the number of years passing. This straightforward calculation helps to determine future values without requiring complex computations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They allow us to translate real-world situations into a mathematical form. This translation is crucial because it enables the solution of problems concerning various quantities.
In our exercise, the expression used is part of a linear equation, which is one of the simplest forms of algebraic expressions. Here, the expression involves constant values (like $154,000, the initial value of the product), variables (such as 't', which stands for the number of years), and operations (addition and multiplication).
Algebraic expressions in the form of linear equations like this can represent real-world scenarios where change happens at a constant rate. This is why they are used in everyday calculations, such as business growth, financial projections, and other instances where consistent per-unit changes occur.
Understanding how to construct and interpret these expressions allows you to apply them to various fields, helping predict outcomes based on set patterns and trends. It is essential for conceptualizing how different elements in a problem interact.

Variable Representation

Variables are symbols used to represent unknown values or quantities that can change. In algebra, variables are usually letters from the alphabet, and they are vital to expressing equations and formulas. They make equations flexible and applicable to numerous situations.
In the provided exercise, the variable 't' is used to represent the number of years after 2005. This means if you want to find the product's value in any future year, you just substitute 't' with the correct numerical representation (e.g., for 2010, 't' equals 10).
Variables allow for dynamic and versatile problem-solving. By using variables, equations become tools that can adapt to various conditions without needing a complete rewrite. They offer a way to generalize a solution, making it universally applicable.
Variable representation keeps equations concise and easy to manage. It ensures that no matter how complex the problem may seem, by manipulating variables, a clear path to the solution is established. It's the foundation that supports exploring different what-if scenarios and testing outcomes efficiently.