/*! 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 12 For exercises \(5-48\), simplify... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 12

For exercises \(5-48\), simplify. $$ \frac{6 w}{w+2}+\frac{w}{w+2} $$

Short Answer

Expert verified
\[\frac{7w}{w + 2}\]

Step by step solution

01

- Identify the common denominator

Both terms in the expression \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( \ \ \ \ \ \ \ \ \ \ \ \ 6w\ \ \ \ ~( \ ( \ w \ \ ) ) +\ ~ \ w\~%(\ w \ + \ \2 \ \ \)\~\ \ \ \ \ \ \ \ \ \ \~) \- \ \ \( \ ( \ \ w ) + 6 )share the same common denominator, which is \w +\2 \.\
02

- Combine the numerators

Since the denominators are identical, combine the numerators as follows:\[\frac{6w + w}{w + 2} \]
03

- Simplify the numerator

Add the coefficients of \(w\) in the numerator to obtain:\[\frac{7w}{w + 2}\]

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
In algebra, when working with fractions, having a common denominator is essential for addition and subtraction. The denominator is the bottom part of the fraction, which represents the total number of equal parts. If the denominators are the same, we can combine the fractions easily. In the exercise \(\frac{6w}{w+2} + \frac{w}{w+2}\), both fractions have the same denominator, \((w+2)\). This common denominator lets us add the fractions directly by combining their numerators while keeping the denominator the same.
Combining Numerators
Once we have a common denominator, the next step is to combine the numerators. The numerator is the top part of the fraction, which represents the portion of the total. For the given exercise, after identifying the common denominator \((w+2)\), we combine the numerators: \(\frac{6w}{w+2} + \frac{w}{w+2}\). To do this, we add the numerators together: \(\frac{6w + w}{w+2}\). This method works because the fractions share the same base (denominator).
Simplifying Fractions
Simplifying fractions helps to present them in their simplest form. After combining the numerators in the exercise, we get \(\frac{6w + w}{w+2}\). The next step is to simplify the numerator. Combine like terms by adding the coefficients of \(w\), which results in \(\frac{7w}{w+2}\). This fraction is already simplified because \(7w\) and \(w+2\) share no common factors. Simplifying fractions like this makes them easier to understand and work with in further calculations.
Elementary Algebra
Elementary algebra involves fundamental concepts and operations, which serve as the building blocks for more advanced mathematics. In our exercise, we used basic algebraic principles to simplify an expression: identifying a common denominator, combining numerators, and simplifying fractions. These steps are core techniques in algebra. Understanding how to manipulate algebraic expressions is crucial for solving equations, understanding functions, and addressing more complicated mathematical problems in the future. Always practice these steps to solidify your foundational algebra skills.

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

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).

For exercises 43-58, (a) solve. (b) check. $$ \frac{9}{10} v+\frac{1}{3}=-\frac{22}{15} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=4\).

For a fixed length of household copper wire, the relationship of the cross- sectional area, \(x\), and the resistance, \(y\), is an inverse variation. When the cross-sectional area is \(3.14 \times 10^{-6} \mathrm{~m}^{2}\), the resistance is \(5.4 \times 10^{-3} \mathrm{ohm}\). a. Find the constant of proportionality, \(k\). Use scientific notation. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the resistance when the cross-sectional area is \(2.05 \times 10^{-6} \mathrm{~m}^{2}\). Round the mantissa to the nearest tenth.

For exercises 1-10, (a) solve. (b) check. $$ \frac{3}{5} x-\frac{1}{4}=\frac{9}{10} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.