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Chapter 1: Problem 5

Solve for the indicated variable in each formula. \(\frac{f}{g}=\frac{h}{k}\) solve for \(k\)

Short Answer

Expert verified
\(k = \frac{g \cdot h}{f}\)

Step by step solution

01

Set up the equation

We are given the equation \(\frac{f}{g}=\frac{h}{k}\). Our goal is to solve for \(k\).
02

Cross-multiply to eliminate fractions

To eliminate the fractions in the equation \(\frac{f}{g}=\frac{h}{k}\), we cross-multiply to get:\(f \cdot k = g \cdot h\).
03

Isolate k

To solve for \(k\), divide both sides of the equation by \(f\):\[k = \frac{g \cdot h}{f}\]

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Key Concepts

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

Cross-Multiplication
When faced with an equation involving two fractions that are equal, cross-multiplication is a powerful technique we can use. Think of a seesaw with two pans on either side. Each pan represents a fraction. The point is to find balance, and cross-multiplying helps us tip the balances equally on both sides by multiplying across the diagonal.
  • The equation \( \frac{f}{g} = \frac{h}{k} \) shows two fractions equal to each other.
  • To cross-multiply, we take \( f \) from the first fraction, and \( k \) from the second, and multiply them together.
  • Similarly, multiply \( g \) and \( h \).
This gives us the equation: \[ f \cdot k = g \cdot h \]Cross-multiplication helps eliminate the fractions, making it easier to handle the equation and solve for unknown quantities.
Solving Equations
Solving equations is like solving a puzzle, where finding the missing piece completes the picture. In mathematics, this missing piece is represented by a variable. Once we cross-multiply and simplify fractions, our equation becomes more straightforward.
For example, the equation after cross-multiplying from \( \frac{f}{g} = \frac{h}{k} \) is \( f \cdot k = g \cdot h \).
  • With the fractions gone, focus on solving an equation involving multiplication.
  • The goal is to manipulate the equation so the variable stands alone.
Every proper step taken brings us closer to the solution. Solving equations requires carefully planned maneuvers to correctly isolate the desired variable.
Variable Isolation
Variable isolation is the process used to get a variable on one side of the equation by itself. In this context, solving for \( k \), is our aim. The process can be thought of as peeling layers away until the variable is left standing alone.
  • Given the equation: \( f \cdot k = g \cdot h \)
  • To isolate \( k \), we want it to be alone on one side, such as the left side.
  • To achieve this, divide both sides of the equation by \( f \).
This division gives: \[ k = \frac{g \cdot h}{f} \]By following these logical steps, you can focus on, and solve for, any variable necessary. Isolation is pivotal in making the variable the star of the equation.

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Most popular questions from this chapter

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\frac{\left(1 \times 10^{-3}\right)\left(3.28 \times 10^{6}\right)}{4 \times 10^{7}}$$

Find all values of \(x\) that make the equation true. $$\frac{1}{2}\left|x-\frac{2}{3}\right|=5$$

Solve the inequality. $$7-3 x<-3$$

Rationalize the numerator, simplifying if possible. $$\frac{\sqrt{x}-\sqrt{y}}{x-y}$$

Perform the indicated operation and simplify. $$\frac{1}{(x-3)^{2}}-\frac{3}{x-3}$$
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