/*! 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 51 Write each rational expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 51

Write each rational expression in lowest terms. $$ \frac{7-b}{b-7} $$

Short Answer

Expert verified
-1

Step by step solution

01

Identify the Rational Expression

The given rational expression is \ \( \frac{7-b}{b-7} \ \).
02

Simplify the Expression

Notice that the numerator \ \( 7 - b \ \) can be rewritten by factoring out \ \( -1 \ \): \ \( 7 - b = -1 \cdot (b - 7) \ \).
03

Substitute the Expression

Substitute \ \( 7 - b \ \) in the numerator with \ \( -1 \cdot (b - 7) \ \): \ \ \ \ \ \ \( \frac{-1 \cdot (b - 7)}{b-7} \ \).
04

Cancel Common Factors

Since \ \( b-7 \ \) is a common factor in both the numerator and the denominator, they cancel each other out, leaving: \ \ \ \ \ \( \frac{-1}{1} \ = -1 \ \).

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.

factoring
Factoring is a method used to break down complex expressions into simpler parts. This is helpful when simplifying rational expressions. For example, if you have an expression like \( 7 - b \), you can factor out \( -1 \) to make it easier to work with. So, \( 7 - b \) can be rewritten as \( -1 \times (b - 7) \). Factoring helps you identify and simplify common factors in both the numerator and denominator, making the overall expression simpler. Always look for the greatest common factor (GCF) when factoring.
rational expressions
Rational expressions are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying these expressions makes them easier to work with. The rational expression in our example is \( \frac{7-b}{b-7} \). To simplify it, we need to manipulate both parts to find common factors that can be canceled. Understanding rational expressions is crucial, as it helps in solving more complex algebraic problems.
canceling common factors
Canceling common factors is a key step in simplifying rational expressions. After you factor both the numerator and the denominator, you may notice common terms that can be canceled out. In our example \( \frac{-1 \times (b-7)}{b-7} \), the \( b-7 \) term appears in both the numerator and the denominator. By canceling this common factor, we are left with \( \frac{-1}{1} \) which simplifies to \( -1 \).
numerator and denominator
The numerator and denominator are the top and bottom parts of a fraction, respectively. When working with rational expressions, it's important to understand their behavior. In our example \( \frac{7-b}{b-7} \), the numerator is \( 7 - b \) and the denominator is \( b - 7 \). By factoring and simplifying, we can reduce the complexity of these parts. Always aim to factor each part to find common factors for cancellation, making the overall expression simpler to understand and work with.

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

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{8}{(3 r-1)^{2}}+\frac{2}{3 r-1}-6 $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3 a}{a+1}+\frac{2 a}{a-3} $$

Concept Check Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." What is meant by the constant of variation in a direct variation problem? If you were to graph the linear equation \(y=k x\) for some nonnegative constant \(k,\) what role would \(k\) play in the graph?

Multiply or divide as indicated. $$ \left(y^{2}-4\right) \div \frac{2-y}{8 y} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3 y}{y^{2}+y z-2 z^{2}}+\frac{4 y-1}{y^{2}-z^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.