/*! 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 20 Solve each equation. See Example... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{2}{9} x-\frac{1}{3}=1 $$

Short Answer

Expert verified
The solution is \( x = 6 \).

Step by step solution

01

Clear Fractions

Multiply every term in the equation by 9 to eliminate the fractions: \[ 9 \cdot \left( \frac{2}{9}x \right) - 9 \cdot \left( \frac{1}{3} \right) = 9 \cdot 1 \]This simplifies to:\[ 2x - 3 = 9 \]
02

Add to Both Sides

Add 3 to both sides of the equation to isolate the term with the variable on one side:\[ 2x - 3 + 3 = 9 + 3 \]This simplifies to:\[ 2x = 12 \]
03

Solve for x

Divide both sides by 2 to solve for \( x \): \[ \frac{2x}{2} = \frac{12}{2} \]This simplifies to:\[ x = 6 \]

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.

Clearing Fractions
When solving linear equations that include fractions, the first step is often to clear these fractions to make the equation simpler to manage. This is done by finding a common denominator or multiplying all terms in the equation by a number that will eliminate the fractions.
In the given example, we have the equation \( \frac{2}{9}x - \frac{1}{3} = 1 \). By multiplying each term by 9, the smallest common multiple of the denominators, we effectively eliminate the fractions. This makes the equation much easier to deal with, as it transforms into \( 2x - 3 = 9 \).
  • Identifying the Common Multiple: Look at all the denominators in the equation.
  • Multiplying Each Term: Multiply every term by the common multiple to clear fractions.
  • Verification: Re-check your resulting equation to ensure no fractions remain!
Isolation of Variables
Once fractions are cleared, the next step is to isolate the variable. This means getting the variable you are solving for on one side of the equation by itself. For the example equation \( 2x - 3 = 9 \), this involves moving all terms that do not contain the variable to the other side of the equation.
To isolate \( x \), you would add 3 to both sides, removing the \( -3 \) from the left side. The equation then becomes \( 2x = 12 \). This process keeps the equation balanced, as whatever you do to one side must be done to the other.
  • Add or Subtract: Use addition or subtraction to move constants from the variable's side.
  • Balance the Equation: Maintain balance by applying changes to both sides.
Algebraic Manipulation
Once the variable is isolated to one side, algebraic manipulation is usually the last step needed to solve the equation. This involves operations such as division or multiplication to finally isolate the variable completely.
In the simplified equation \( 2x = 12 \), the coefficient of \( x \) is 2. To solve for \( x \), divide both sides by 2, giving you \( x = 6 \). This straightforward division isolates \( x \), solving the equation entirely.
  • Divide or Multiply: Remove any coefficients using division or multiplication as needed.
  • Double-Check: Always substitute your solution back into the original equation to verify accuracy.

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

To make an international telephone call, you need the code for the country you are calling. The codes for Belgium, France, and Spain are three consecutive integers whose sum is \(99 .\) Find the code for each country. (Source: The World Almanac and Book of Facts)

The sum of the measures of the angles of a parallelogram is \(360^{\circ} .\) In the parallelogram below, angles \(A\) and \(D\) have the same measure as well as angles \(C\) and \(B\). If the measure of angle \(C\) is twice the measure of angle \(A,\) find the measure of each angle.

Solve each inequality. Write each answer using solution set notation. $$ -6 x+2<-3(x+4) $$

Twice the difference of a number and 8 is equal to three times the sum of the number and 3 . Find the number.

Six more than twice a number is greater than negative fourteen. Find all numbers that make this statement true.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics 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.