Problem 6 Solve each equation for $y$. ... [FREE SOLUTION]

Chapter 6: Problem 6

Solve each equation for $y$. $$\frac{1}{n}=\frac{a}{y}-\frac{w}{a}$$

Short Answer

Expert verified

y = \frac{a^2n}{a + wn}

Step by step solution

Isolate the term with the variable y

Rewrite the equation to isolate the term with y: $ \frac{1}{n} = \frac{a}{y} - \frac{w}{a} $ Add $ \frac{w}{a} $ to both sides: $ \frac{1}{n} + \frac{w}{a} = \frac{a}{y} $

Find a common denominator

To combine the fractions on the left side, find a common denominator. The least common multiple of n and a is na. Thus, $\frac{1}{n} + \frac{w}{a} = \frac{a}{y} $ becomes: $\frac{a}{an} + \frac{wn}{an} = \frac{a}{y} $

Combine the fractions

Combine the fractions on the left side: $ \frac{a + wn}{an} = \frac{a}{y} $

Cross multiply to solve for y

Cross multiply to get rid of the fractions: $ y(a + wn) = a \times an $ Simplify it: $ y(a + wn) = a^2n $

Solve for y

Finally, divide both sides by $ a + wn $: $ y = \frac{a^2n}{a + wn} $

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Isolating the Variable

When solving equations, you often need to isolate the variable to one side. This helps in focusing on the term you want to solve for. In our example, we have the equation $\frac{1}{n} = \frac{a}{y} - \frac{w}{a}$.
To isolate the term with $y$, add $\frac{w}{a}$ to both sides:
$\frac{1}{n} + \frac{w}{a} = \frac{a}{y}$.
Once the term with the variable is isolated, it is easier to manipulate the equation to solve for $y$.
Isolating the variable is a crucial first step.

Common Denominator

To add or subtract fractions, they need a common denominator. This means finding a shared multiple of the denominators involved. In the equation $\frac{1}{n} + \frac{w}{a} = \frac{a}{y}$, n and a are the denominators.
The least common multiple of n and a is na. Thus, we rewrite the fractions:
$\frac{a}{an} + \frac{wn}{an}$.
Having a common denominator allows us to combine these fractions easily.

Cross Multiplication

Cross multiplication is a technique for solving equations involving two fractions. It helps to eliminate the fractions. For the equation $\frac{a+wn}{an} = \frac{a}{y}$, multiply across the equals sign:
$y(a + wn) = a \times an$.
Cross multiplication transforms the equation into a simpler form that can be manipulated further.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations. After cross-multiplying, we get \y(a + wn) = a^2n\. To solve for $y$, divide both sides by $a + wn$:
$y = \frac{a^2n}{a + wn}$.
This step-by-step rearrangement of terms is key to solving equations accurately.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve each equation for \(y\). $$\frac{1}{n}=\frac{a}{y}-\frac{w}{a}$$

Short Answer

Step by step solution

Isolate the term with the variable y

Find a common denominator

Combine the fractions

Cross multiply to solve for y

Solve for y

Key Concepts

Isolating the Variable

Common Denominator

Cross Multiplication

Algebraic Manipulation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.