Problem 77 - Algebra A Combined Function

Question

Solve. \(N=R+\frac{V}{G}\) for \(V\) (Urban forestry: tree plantings per year)

Step-by-Step Solution

Answer

\(V = G(N - R)\)

Step 1: Identify What We Need to Solve For

We need to solve the equation for the variable \(V\). This means we have to isolate \(V\) on one side of the equation.

Step 2: Subtract \(R\) from Both Sides

The given equation is \(N = R + \frac{V}{G}\). To isolate \(V\), first subtract \(R\) from both sides to get \(N - R = \frac{V}{G}\).

Step 3: Eliminate the Denominator

To solve for \(V\), we need to eliminate the fraction. Multiply both sides of the equation by \(G\) to give \(G(N - R) = V\).

Step 4: Solution for \(V\)

The variable \(V\) is now isolated. So, \(V = G(N - R)\). This is the solution for \(V\) in terms of \(N\), \(R\), and \(G\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols can represent numbers in equations and formulas, allowing us to solve for unknown quantities. In algebra, equations can range from simple to complex. They are tools used to represent real-world problems, like figuring out how many trees to plant in a year to improve urban forestry. Such problems can be structured using equations, enabling us to find solutions that help in planning and decision-making. Algebraic equations often include:

• Variables, which are symbols that stand in for unknown values.
• Operations, like addition, subtraction, multiplication, and division.
• Constants, which are fixed values used in equations.

Understanding algebra is about recognizing patterns and forming relationships between different quantities. This makes algebra a crucial foundational skill for solving real-life problems involving numbers.

Variable Isolation

Variable isolation refers to the process of solving equations by getting the variable of interest by itself on one side of the equation. This process involves rearranging the equation and performing various operations to isolate the variable. It is a critical step in finding the solution to an equation, particularly in real-world applications. To isolate a variable, you typically:

• Perform inverse operations to "move" terms from one side of the equation to the other.
• Ensure any fractions are cleared, often by multiplying both sides by a common denominator.
• Keep the equation balanced by performing the same operation on both sides.

In the given exercise, the goal was to solve for the variable \(V\). By following steps such as subtracting \(R\) (to get rid of the terms not involving \(V\)) and multiplying by \(G\) (to eliminate the fraction), \(V\) was successfully isolated. This method simplifies the equation, making the unknown variable easy to identify and solve.

Linear Equations

Linear equations are a type of algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations form straight lines when graphed, hence the name 'linear'. They are fundamental in algebra due to their straightforward structure, making them relatively simple to solve.Key characteristics of linear equations include:

• They contain no exponents higher than one, meaning the variables are not squared, cubed, etc.
• They appear in the form \(ax + b = 0\), where \(a\) and \(b\) are constants.
• They're used often in real-world scenarios, such as calculating costs and budgeting.

In the original problem, the equation \(N=R+\frac{V}{G}\) is consider linear because it can be arranged in a form where \(N\), \(R\), and \(V\) are linear terms. By transforming the equation to solve for \(V\), we observe linear manipulation in isolating \(V\), highlighting how understanding linear equations helps solve diverse practical problems.