/*! 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 31 Solve each rational equation. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 31

Solve each rational equation. $$\frac{8 y}{y+1}=4-\frac{8}{y+1}$$

Short Answer

Expert verified
The solution to the equation is \( y = -1 \)

Step by step solution

01

Identifying the common denominator

In this problem, the common denominator across the entire equation is \( (y+1) \).
02

Multiply through by the common denominator

Next, multiply each term of the equation by the common denominator \( (y+1) \). That simplifies the equation to:\n \( 8y = 4(y+1) - 8 \)
03

Distribute and simplify

Next, distribute the 4 in the right side of the equation and simplify to get:\n \( 8y = 4y + 4 - 8 \n 8y = 4y - 4 \)
04

Solve for y

Next, re-arrange the equation to isolate \( y \) on one side:\n \( 8y - 4y = - 4 \n 4y = -4 \n So, \( y = -4 / 4 = -1 \)

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.

Common Denominator
When solving rational equations, finding a common denominator is a crucial step. A rational equation is one that contains fractions with variables in the denominators. For example, consider the equation \( \frac{8y}{y+1} = 4 - \frac{8}{y+1} \). Both fractions here have the same denominator, which is \( (y+1) \).
This denominator allows us to simplify the problem by eliminating fractions, making it easier to solve the equation. Utilizing a common denominator is like speaking the same language for all fractions involved in the equation.

To handle it effectively:
  • Identify a shared denominator among all fractions鈥攈ere it is \( (y+1) \).
  • Multiply every term in the equation by this common denominator, thereby canceling out the denominators.
This step helps to align terms, getting rid of fractions and transforming the problem into a simpler linear equation.
Equation Simplification
Once a common denominator is established and the equation is free from fractions, it's time to simplify the equation further. The goal here is to get the equation into a form that is easy to solve.

Consider the equation: \( 8y = 4(y+1) - 8 \).
To simplify:
  • Distribute any numbers in front of parentheses鈥攈ere, it's 4 on the right side, which gives \( 4y + 4 \).
  • Combine like terms to condense the equation as much as possible鈥擻( 4 - 8 \) simplifies to \( -4 \).

This leads to the simplified equation \( 8y = 4y - 4 \). Every simplification step reduces complexity, turning it into a straightforward algebraic expression and setting the stage for easily finding the variable.
Isolating Variables
The final and most important step is isolating the variable to find its value. This is done by manipulating the equation until the variable is alone on one side.

With the simplified equation \( 8y = 4y - 4 \), follow these steps:
  • Re-order the terms to gather all terms containing the variable on one side. Subtract \( 4y \) from both sides to isolate terms with \( y \): \( 8y - 4y = -4 \).
  • Perform any necessary arithmetic, such as division or subtraction, to further isolate \( y \). Here, divide both sides by 4 to get \( y = \frac{-4}{4} \).
This yields \( y = -1 \). Isolating the variable lets us solve for it directly, revealing the solution to the equation. This step requires careful repositioning of terms to neatly solve for the unknown.

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

If the ratio of the corresponding sides of two similar triangles is 1 to 1 ( \(\frac{1}{1}\) ), what must be true about the triangles?

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

Describe two similarities between the following problems: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Describe how to identify the corresponding sides in similar triangles.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.