/*! 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 1 Solve the following rational equ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 1

Solve the following rational equations. \(\frac{2 x}{5}=\frac{x-14}{6}\)

Short Answer

Expert verified
Answer: The value of x that satisfies the given rational equation is \(x = -10\).

Step by step solution

01

Identify the equation

We are given the rational equation: \(\frac{2x}{5} = \frac{x-14}{6}\)
02

Clear the denominators

In order to clear the denominators, we need to find the least common multiple (LCM) of the two denominators 5 and 6. In this case, the LCM of 5 and 6 is 30. Now, we'll multiply both sides of the equation by the LCM: \(30 \cdot \frac{2x}{5} = 30 \cdot \frac{x-14}{6}\)
03

Simplify the equation

After multiplying both sides by 30, we can simplify the equation: \((30/5) \cdot 2x = (30/6) \cdot (x-14)\) \(12x = 5(x-14)\)
04

Distribute the values

Now we will distribute the 5 to both terms inside the parenthesis: \(12x = 5x - 70\)
05

Isolate x

To isolate x, we will subtract the 5x from both sides: \(7x = -70\) Now, divide both sides by 7 to get the final x-value: \(x = -10\) So, the solution to the rational equation \(\frac{2x}{5} = \frac{x-14}{6}\) is \(x = -10\).

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.

Understanding Least Common Multiple (LCM)
When solving rational equations, one key step is to clear the denominators. To do this, we find the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide without a remainder. For example, in the equation \(\frac{2x}{5} = \frac{x-14}{6}\), the denominators are 5 and 6.
To find the LCM of 5 and 6:
  • List the multiples of each number:
    • Multiples of 5: 5, 10, 15, 20, 25, 30, ...
    • Multiples of 6: 6, 12, 18, 24, 30, ...
  • The smallest common multiple is 30.
By multiplying both sides of the equation by 30, we eliminate the fractions and make the equation easier to solve.
Isolating the Variable
Once the equation is free of fractions, the next step is to isolate the variable, which means getting the variable by itself on one side of the equation. In the example \(12x = 5(x-14)\), we first distribute the 5 to the terms inside the parenthesis.
This gives us:
  • \(12x = 5x - 70\)
To isolate \(x\), we subtract \(5x\) from both sides:
  • \(12x - 5x = -70\)
  • \(7x = -70\)
Finally, divide both sides by 7 to solve for \(x\):
  • \(x = -10\)
This step-by-step approach allows us to find the value of the variable.
Simplifying Equations
Simplifying equations involves making them as straightforward as possible, often by combining like terms and eliminating complex fractions. After using the LCM to clear fractions, we'll apply simplification techniques.
In our example, we simplified the equation:
  • After multiplying both sides by 30, we ended up with \(12x = 5(x-14)\).
  • Distribution led to \(12x = 5x - 70\).
Next, combine like terms:
  • Subtract \(5x\) from \(12x\) to simplify the left side.
  • Combine to get \(7x = -70\).
Divide by 7 to find \(x = -10\).
Simplification helps to solve equations more easily by reducing clutter and focusing only on the necessary steps.

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

For the following problems, perform the indicated operations. $$ \frac{6 a+5}{(2 a+1)(4 a-3)}+\frac{4 a+1}{2 a+1} $$

For the following problems, divide the polynomials. $$ 3 r^{2}-17 r-27 \text { by } r-7 $$

For the following problems, divide the polynomials. $$ y^{3}-2 y^{2}-49 y-6 \text { by } y+6 $$

For the following problems, find the solution. When one third of a number is added to the reciprocal of the number, the result is \(\frac{-127}{90} .\) What is the number?

For the following problems, perform the indicated operations. $$ \frac{x^{2}}{18} \cdot \frac{3}{x^{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.