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Chapter 1: Problem 42

Solve the equation for the indicated variable. $$ w=\frac{k u v}{s^{2}} ; u $$

Short Answer

Expert verified
The equation solved for \(u\) is: \(u = \frac{s^2w}{kv}\).

Step by step solution

01

Identify the equation

The given equation is: $$ w = \frac{kuv}{s^2} $$ Our goal is to find the value of \(u\) in terms of the other variables.
02

Cross-multiply to remove the fraction

To eliminate the fraction, multiply both sides of the equation by \(s^2\): $$ s^2w = kuv $$
03

Isolate u

In order to isolate \(u\), we have to divide both sides of the equation by \(kv\): $$ u = \frac{s^2w}{kv} $$
04

Final Answer

The equation solved for \(u\) is: $$ u = \frac{s^2w}{kv} $$

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Key Concepts

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

Cross Multiplication
Cross multiplication is a useful technique in algebra especially when dealing with fractions. It involves multiplying across the equal sign so that each side of the equation has a product. This technique helps in clearing fractions by multiplying each term by the common denominator.

For example, in the equation \( w = \frac{kuv}{s^2} \), cross multiplying by \( s^2 \) eliminates the fraction and simplifies the equation to \( s^2w = kuv \). With cross multiplication, the key is to ensure you multiply every term in the equation by the same value to keep the equation balanced and true.

  • Remember to always multiply every term to avoid errors.
  • This method prominently applies to proportions but is also useful in other equation types to eliminate fractions.
  • After cross multiplying, simplify the equation as needed before proceeding with further steps.
Isolating Variables
Isolating variables is a fundamental aspect of solving algebraic equations. The goal is to get the variable of interest alone on one side of the equation, which allows us to solve for its value in terms of other variables.

In our example, we want to solve for \( u \) using the equation \( s^2w = kuv \). To isolate \( u \) we must eliminate every other term attached to \( u \). Since \( u \) is being multiplied by \( kv \) in our equation, we can do this by dividing both sides of the equation by \( kv \) which yields \( u = \frac{s^2w}{kv} \).

  • Always perform the same operation on both sides of the equation to maintain the balance.
  • Be thorough in simplifying the isolated variable expression to its simplest form for the answer.
  • Isolation is also critical when you have equations with multiple variables and you need to express one variable in terms of the others.
Algebraic Manipulation
Algebraic manipulation encompasses a variety of techniques used to rearrange and simplify equations. The goal of these techniques is to solve for a variable or to express the equation in a more usable form. It includes skills like distribution, combining like terms, and using inverse operations.

In the case of the example provided, once we have cross multiplied to remove the fraction and rearranged the equation to isolate \( u \), the expression \( u = \frac{s^2w}{kv} \) is a result of algebraic manipulation. Here, manipulation involves the distribution and removal of terms associated with \( u \) to solve for it explicitly.

  • Mastering algebraic manipulation is key for problem solving in algebra.
  • Remember to keep your work neat and stepwise to avoid confusion.
  • Understanding the properties of equality and the various operations is vital for effective algebraic manipulation.

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Most popular questions from this chapter

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