/*! 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 65 In Exercises 65-68, solve the ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 65

In Exercises 65-68, solve the rational equation. $$ \frac{x}{5}+\frac{2 x}{5}=3 $$

Short Answer

Expert verified
The solution to the rational equation is \(x = 5\).

Step by step solution

01

Combine Similar Fractions

The first step is to combine like terms on the left side of the equation, producing \(\frac{3x}{5} = 3\).
02

Multiply by Common Denominator

Next step is to get rid of the fractions, which can be achieved by multiplying both sides of the equation by the common denominator, 5.\nDoing so, results in the equation \(3x = 15\).
03

Solve for the Variable

To solve for \(x\), we must isolate it to one side of the equation. We achieve this by dividing both sides of the equation by 3, resulting in \(x = 5\).

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.

Combining Like Terms
When solving rational equations, the first important step often involves combining like terms. This means finding terms that have the same variable and simplifying them as a single term.
In the given rational equation \(\frac{x}{5} + \frac{2x}{5} = 3\), we notice that both terms on the left contain the variable \(x\) with the same denominator 5.
  • Since the denominators are identical, these terms are "like" terms.
  • To combine them, simply add their numerators together.
This gives us \(\frac{1x + 2x}{5} = \frac{3x}{5}\), effectively reducing the equation to a simpler form, \(\frac{3x}{5} = 3\).
By combining like terms, we make the equation easier to manipulate in subsequent steps.
Common Denominator
Rational equations often involve fractions, and dealing with them effectively requires finding a common denominator.
In our case, after combining like terms, we are left with \(\frac{3x}{5} = 3\).
  • The common denominator, in this case, is 5, which we use to eliminate the fraction.
  • This process transforms the equation into a simpler linear equation, making it easier to solve for the variable.
Multiply every term in the equation by 5, the least common multiple of the denominators, to eliminate the fractions:\[ 3x = 15 \] Removing the fractions makes the equation straightforward, setting the stage for the final step of solving for the variable.
Solving for Variables
Once you have simplified the equation and eliminated fractions, the next step is solving for the variable.
We simplified our rational equation to \(3x = 15\).
  • The goal now is to isolate the variable \(x\) on one side of the equation.
  • To achieve this, divide both sides by the coefficient of \(x\) (which is 3).
By performing this division:\[ \frac{3x}{3} = \frac{15}{3} \] You solve for \(x\), finding that \(x = 5\).
The equation is now solved, and through isolating the variable, we have successfully found the solution.

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

What is a verbal model of a real-life problem?

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=0.55 x+40,000 \quad R=0.85 x $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{lr} y \geq & -6 \\ y \leq & -8 x+9 \\ x \geq & 0 \\ y \leq & 0 \end{array}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y>2 x \\ y>-x+4 \end{array}\right. $$

In Exercises \(63-66\), sketch the graph of the equation. $$ 2 x+4 y=8 $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure 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.