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Chapter 5: Problem 71

Solve each equation for the specified variable. $$ k=d F-D F \text { for } F $$

Short Answer

Expert verified
F = \frac{k}{d - D}

Step by step solution

01

Identify the equation

The given equation is \( k = dF - DF \). The goal is to solve for the variable \( F \).
02

Factor out the common term

Notice that \( F \) is the common factor in the terms on the right side of the equation. Factor \( F \) out: \[ k = F(d - D) \].
03

Isolate the variable F

To isolate \( F \), divide both sides of the equation by \( d - D \): \[ F = \frac{k}{d - D} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a key method used to simplify algebraic expressions and solve equations. In this context, factoring means identifying a common term across multiple parts of the equation. For instance in the equation, \( k = dF - DF \), we noticed that the variable \( F \) appears in both terms on the right. By factoring, we pull out the common factor, in this case, \( F \), and rewrite it as \( F(d - D) \). This simplifies the equation and prepares it for easier manipulation in later steps.
Isolating Variables
Isolating variables is a technique used to single out the variable you are trying to solve. In the equation \( k = F(d - D) \), our goal is to solve for \( F \). To isolate \( F \), we need to move everything else to the other side of the equation. We do this by performing algebraic operations that simplify or cancel out other terms surrounding \( F \). Here, we divide both sides of the equation by \( d - D \), which leaves \( F \) on its own: \( F = \frac{k}{d - D} \). Now, \( F \) is isolated and we have our solution.
Algebraic Manipulation
Algebraic manipulation involves reshaping an equation into a more useful or simpler form using various algebraic techniques. This can include operations like addition, subtraction, multiplication, division, and factoring. In our exercise, we employed algebraic manipulation first by factoring to simplify the expression and then by dividing to isolate the variable. These steps must be performed carefully to maintain the equation鈥檚 balance. It鈥檚 important to make sure that any operation done on one side of the equation is also done on the other side to keep the equation true.

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Most popular questions from this chapter

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