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Linear equations are the foundation of many algebraic problems. In the case of our revenue function, the equation is expressed as: \( R = 17.5x + 138.5 \). This is a linear equation because it has the form \( y = mx + b \), where: - \( R \) (or \( y \)) represents the dependent variable (total revenue). - \( 17.5 \) (or \( m \)) is the slope of the line; it shows the rate at which revenue changes per year. - \( x \) is the independent variable (number of years since 2005). - \( 138.5 \) (or \( b \)) is the y-intercept; the revenue amount when \( x = 0 \) (i.e., in 2005). Linear equations are crucial in modeling real-world scenarios like revenue, as they provide a predictable pattern based on given variables.

When you encounter an algebraic equation, one of the main goals is to solve for the unknown variable. In this problem, determining the value of \( x \) (years since 2005) or \( R \) (revenue) can provide significant insights. Let's break this down: - **Determining Revenue for a Specific Year**: To find the revenue for a particular year (e.g., 2007), substitute that year (converted to \( x \)) into the equation. In our example: - 2007 corresponds to \( x = 2 \) - Substitute \( x = 2 \) into the equation and solve for \( R \). - **Finding the Year for a Given Revenue**: To know when revenue reaches a specific amount (e.g., \$226 billion), we set \( R \) to that value and solve for \( x \): - Substitute \( R = 226 \) into the equation and isolate \( x \) to find the number of years since 2005. Solving for variables provides the specific information necessary to understand and predict values in a given context.

The substitution method is a handy algebraic tool for solving equations. Here's how it works with our revenue example: 1. **Identify the Given Information**: Here, the function is \( R = 17.5x + 138.5 \). 2. **Substitute the Known Value**: For part (a), we want to find revenue for 2007: - Calculate \( x = 2007 - 2005 = 2 \). - Substitute \( x = 2 \) into the equation: \( R = 17.5(2) + 138.5 \). 3. **Solve the Equation**: Perform the arithmetic to find \( R \). The same method applies when solving for \( x \) given a specific revenue value. It's about replacing the variable with known values and solving the simplified equation. This technique makes finding specific answers straightforward and reliable.

Algebraic modeling involves creating equations to represent real-world situations. In our example, the equation \( R = 17.5x + 138.5 \) models the relationship between time and revenue for telecommunications. - **Why Use Algebraic Modeling?** - It helps predict future values based on current or past data. - You can determine unknown variables affecting the situation (e.g., how long until revenue hits a certain amount). - **Creating Models**: - Identify the relationship you want to model (revenue over time). - Choose appropriate variables (\( x \) for years since 2005, \( R \) for revenue). - Determine the equation form (linear relationship here). - **Benefits**: - Provides a clear, mathematical way to analyze trends and make forecasts. - Helps visualize how changes in one variable affect another. By effectively applying algebraic modeling, we can simplify complex scenarios and make informed decisions based on predicted outcomes.