Problem 49 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( G = \frac{V}{N - R} \)

Step 1: Identify the Equation

The given equation is \( N = R + \frac{V}{G} \). We need to solve for the variable \( G \).

Step 2: Isolate the Fraction Term

Subtract \( R \) from both sides of the equation to isolate the fraction: \( N - R = \frac{V}{G} \).

Step 3: Clear the Fraction

To solve for \( G \), multiply both sides by \( G \): \( G(N - R) = V \).

Step 4: Solve for G

Divide both sides by \( (N - R) \) to get \( G \) alone: \( G = \frac{V}{N - R} \).

Key Concepts

Equation Solving

Equation solving is a foundational concept in algebra that involves finding the value of a variable that makes an equation true. In any equation, you typically have an expression set equal to another expression. The goal is to manipulate the equation using algebraic operations to solve for the unknown.There are various methods for solving equations depending on the type of equation. These can include adding, subtracting, multiplying, or dividing both sides by the same number. By doing this, you maintain the balance of the equation. In our given problem, the equation \(N = R + \frac{V}{G}\) needs to be solved for \(G\).General steps for solving equations include:

• Identifying and understanding the equation.
• Selecting appropriate operations to simplify or rearrange the equation.
• Transposing terms to isolate the variable of interest.

Once the variable is isolated, you can often read off its value directly. Patience and practice are key in mastering equation solving.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. It means rearranging the equation so that one variable stands alone on one side of the equation. This process often involves manipulating the equation with inverse operations to 'undo' whatever operations are currently being applied to the variable.In the exercise, our goal was to isolate \(G\). Starting with the equation \(N = R + \frac{V}{G}\), we first needed to move terms around to separate \(G\). Subtracting \(R\) from both sides gave us \(N - R = \frac{V}{G}\), effectively isolating the fraction containing \(G\).Main steps to isolate a variable:

• Identify the variable you need to solve for.
• Use algebraic operations to "undo" additions, subtractions, multiplications, etc.
• Be consistent with operations, doing the same on both sides of the equation.

Consistently applying these steps will lead you to the solution. It requires careful attention to ensure that balance in the equation is maintained.

Rational Equations

Rational equations are equations that involve rational expressions, which are fractions containing variables in the numerator, the denominator, or both. Solving these equations often involves clearing the fractions to simplify the process of finding the solution.In our problem, \(N = R + \frac{V}{G}\), the term \(\frac{V}{G}\) is a rational expression. A key strategy in solving rational equations is to eliminate the fractional part early in the process. This can often be achieved by multiplying both sides of the equation by the denominator. Here, multiplying both sides by \(G\) removed the fraction, transforming the equation to \(G(N - R) = V\).When working with rational equations:

• It's important to identify the fractions and their denominators.
• Clear the fractions by multiplying all terms by the least common denominator (LCD) if needed.
• Solve the resulting equation, which should now be easier to handle.

The focus is on making the equation more manageable by simplifying its structure, thereby making it easier to isolate variables and find solutions.