Problem 50 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ \bar{v}=\frac{1}{2}\left(v+v_{0}\right) \quad \text { for } v_{0} $$

Step-by-Step Solution

Answer

\( v_{0} = 2\bar{v} - v \)

Step 1: Understand the formula

The equation we need to manipulate is given by \( \bar{v}=\frac{1}{2}(v+v_{0}) \). We are tasked to solve for \( v_{0} \). This means we need to isolate \( v_{0} \) on one side of the equation.

Step 2: Eliminate the fraction

To eliminate the fraction, multiply both sides of the equation by 2: \[ 2\bar{v} = v + v_{0} \]. This step clears the fraction by multiplying everything by the denominator.

Step 3: Isolate \( v_{0} \)

We now need \( v_{0} \) to be all by itself. Subtract \( v \) from both sides: \[ v_{0} = 2\bar{v} - v \]. This step has isolated \( v_{0} \) completely.

Key Concepts

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging an equation to make it easier to work with or to solve for a particular variable. This often involves using properties of numbers and operations such as addition, subtraction, multiplication, and division.
In the given exercise, the formula is \(\bar{v} = \frac{1}{2}(v + v_0)\).Here, manipulating the equation algebraically allows us to solve for \(v_0\).

• We use multiplication to clear the fraction by multiplying the entire equation by 2.
• Manipulating equations like this is fundamental for simplifying and solving them properly.
• Careful attention to each operation ensures that each change is correct and controlled.

Practicing algebraic manipulation with different equations helps to solidify understanding and improve problem-solving skills.

Isolating Variables

Isolating variables means arranging the equation in such a way that the desired variable is by itself on one side of the equation. In this exercise, we want to isolate \(v_0\).
Starting with \(2\bar{v} = v + v_0\), we take these steps:

• Look at all the terms on both sides of the equation.
• Understand the operations that attach each term to the variable \(v_0\).
• Subtract \(v\) from both sides to get \(v_0\) alone, resulting in \(v_0 = 2\bar{v} - v\).

This process requires keen attention and understanding of how each term affects the variable you want to isolate.
With practice, isolating variables becomes a straightforward and intuitive task.

Fraction Elimination

Fraction elimination is often necessary to simplify equations and make the terms easier to handle.It involves getting rid of fractions by multiplying by the reciprocal of the fraction's denominator.
In the equation \(\bar{v} = \frac{1}{2}(v + v_0)\),we eliminate the fraction by multiplying both sides by 2.
This gives us:\[2\bar{v} = v + v_0\]

• Find the common denominator or the denominator you need to cancel.
• Multiply the whole equation by this number to remove the fraction.
• Ensure that you apply this multiplication to every term in the equation.

Eliminating fractions simplifies the manipulation process and helps in moving forward efficiently with isolation and solving.