/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 Solve formula for the specified ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 89

Solve formula for the specified variable. \(\frac{1}{x}=\frac{1}{y}-\frac{1}{z}\) for \(y\)

Short Answer

Expert verified
y = \frac{xz}{x + z}

Step by step solution

01

Combine the fractions on the right-hand side

To combine the fractions on the right-hand side, find a common denominator. For the fractions \(\frac{1}{y} - \frac{1}{z} \), the common denominator will be \(yz\). Therefore, rewrite the right-hand side expression: \[ \frac{1}{y} - \frac{1}{z} = \frac{z - y}{yz} \]
02

Set up the equation

Rewrite the equation using the common denominator found in Step 1: \[ \frac{1}{x} = \frac{z - y}{yz} \]
03

Cross-multiply to clear the fractions

Cross-multiply to eliminate the fractions: \[ yz = x(z - y) \]
04

Solve for y

Distribute x in the equation: \[ yz = xz - xy \] To isolate y, move all terms involving y to one side: \[ yz + xy = xz \] Factor out y from the left-hand side: \[ y(z + x) = xz \] Finally, solve for y by dividing both sides by (z + x): \[ y = \frac{xz}{x + z} \]

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.

Cross-Multiplication
Cross-multiplication is a method used to solve equations involving fractions by eliminating the denominators. This makes the equations easier to work with. When you have an equation of the form \[ \frac{a}{b} = \frac{c}{d} \], you can cross-multiply by multiplying the numerator of one fraction by the denominator of the other, and vice versa. This gives us \[ a \times d = b \times c \]. Performing cross-multiplication in the exercise, we start with the equation \[ \frac{1}{x} = \frac{z - y}{yz} \]. Cross-multiplying both sides, we get \[ yz = x(z - y) \]. This transforms the equation into a linear form without fractions, which can be more straightforward to solve.
Common Denominators
Finding common denominators is essential when adding, subtracting, or comparing fractions. A common denominator for two or more fractions is a number that all denominators can divide evenly. For \[ \frac{1}{y} \] and \[ \frac{1}{z} \], the common denominator is \[ yz \], because both y and z can multiply to form this denominator. By converting the fractions to have this common denominator, you can combine them into a single fraction. In the exercise, we perform this step to rewrite \[ \frac{1}{y} - \frac{1}{z} \] as \[ \frac{z - y}{yz} \]. This turns a subtraction problem into a simpler fraction, making the next steps in solving the equation much more manageable.
Factoring
Factoring is the process of breaking down an expression into simpler expressions (factors) that multiply together to give the original expression. This technique is vital for solving polynomial equations. In the exercise, after cross-multiplying, we reach the equation \[ yz = x(z - y) \]. To isolate the variable y, we rearrange terms to get \[ yz + xy = xz \]. By factoring y out of the left-hand side of the equation, we obtain \[ y(z + x) = xz \]. Finally, solving for y involves dividing both sides by \[ z + x \], resulting in \[ y = \frac{xz}{x + z} \]. This step simplifies the expression of y in terms of x and z, concluding the solution process.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{x^{2} y+y+x^{2} z+z}{x y+x z} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{1-8 r^{3}}{8 r^{2}+4 r+2} $$

Multiply. Write each answer in lowest terms. $$ \frac{3 x}{x+3} \cdot \frac{(x+3)^{2}}{6 x^{2}} $$

Multiply. Write each answer in lowest terms. $$ \frac{12 m^{5}}{-2 m^{2}} \cdot \frac{6 m^{6}}{28 m^{3}} $$

Write four equivalent forms for each rational expression. $$ -\frac{2 x-9}{7 x-1} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.