Problem 78 - Algebra A Combined Function

Question

Solve. B=\frac{F}{P-V} \text { for } V

Step-by-Step Solution

Answer

The solution for \( V \) is \( V = \frac{F - BP}{B} \).

Step 1: Understand the given formula

You are given the formula \( B = \frac{F}{P-V} \) and you need to solve for \( V \). This means rearranging the equation to isolate \( V \) on one side.

Step 2: Clear the fraction

Multiply both sides of the equation by \( (P - V) \) to eliminate the fraction: \( B(P - V) = F \).

Step 3: Distribute B

Distribute the \( B \) on the left side of the equation: \( BP - BV = F \).

Step 4: Isolate terms involving V

To isolate terms involving \( V \), subtract \( BP \) from both sides: \( -BV = F - BP \).

Step 5: Solve for V

Divide both sides of the equation by \( -B \) to solve for \( V \): \( V = \frac{BP - F}{B} \).

Step 6: Simplify the expression

Since \( (-1) \) is a constant, the negative sign can be moved: \( V = \frac{F - BP}{B} \).

Key Concepts

Solving for a Variable

When solving for a variable, you are working to isolate that particular variable on one side of the equation. This means you aim to have it all alone, making it the subject of the formula. In the exercise, we are tasked with the variable \( V \) in the equation \( B = \frac{F}{P-V} \).
To solve for \( V \), the main goal is to move all other terms and variables to the opposite side of the equation, leaving \( V \) isolated. This step-by-step methodical approach requires:

• Understanding the given formula
• Recognizing the need to rearrange it
• Moving terms around logically to isolate \( V \)

Throughout the process, you apply basic algebraic operations—like multiplying or dividing throughout the equation—until \( V \) stands alone on one side.

Rearranging Formulas

Rearranging formulas is a critical process in algebra, allowing us to solve equations more easily by isolating a desired variable. It entails logically shifting terms from one side of an equation to the other, keeping the equation balanced.
The given equation, \( B = \frac{F}{P-V} \), is rearranged by eliminating the fraction and rearranging terms to bring \( V \) alone on one side. Here are the steps:

• Multiply both sides by \( (P-V) \) to bring \( V \) from the fraction to the main equation.
• Distribute any coefficients like \( B \) across terms if necessary.
• Move entire expressions (like \( BP \)) across the equals sign.
• Do the inverse operation (adding instead of subtracting or vice versa) to maintain balance.

By performing these steps, you reorganize the formula neatly, enabling you to see and solve it more clearly.

Fraction Elimination

Fraction elimination is a valuable technique in algebra to simplify equations, especially when aiming to solve for specific variables. Fractions can make equations look more complex, so removing them often makes the equation easier to handle.
In our specific problem, the equation \( B = \frac{F}{P-V} \) contains a fraction that we need to eliminate. Here’s how it can be done:

• Multiply both sides of the equation by the denominator \( (P-V) \) to clear the fraction and simplify the equation.
• This action shifts the fraction into a standard linear equation, allowing the variable \( V \) to stay with other terms as seen in \( B(P-V) = F\).

Once fractions are removed, the path to solving for variables becomes more straightforward and less prone to arithmetic errors.

Algebraic Manipulation

Algebraic manipulation refers to the variety of techniques to rearrange, simplify, and solve equations. This includes employing addition, subtraction, multiplication, and division strategically to both sides of an equation.
In the problem \( B = \frac{F}{P-V} \), we performed several manipulations:

• Distribution, where \( B(P - V) \) becomes \( BP - BV \).
• Rearranging terms by moving \( BP \) to the other side to isolate terms with \( V \).
• Dividing by \(-B\) to solve for \( V \) and simplify the equation \( V = \frac{F - BP}{B} \).

These methods are integral to algebraic problem solving, allowing us to move components of the equation freely until we isolate the variable needed.