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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

Eliminate Fractions by Finding a Common Denominator

The equation \( \frac{2}{9}x - \frac{1}{3} = 1 \) contains fractions. To eliminate them, find the least common denominator, which is 9 in this case. Multiply every term by 9 to get \( 2x - 3 = 9 \).
02

Move the Constant to the Other Side

Add 3 to both sides of the equation to isolate the term containing \( x \). This yields \( 2x = 12 \).
03

Solve for x

Divide both sides by 2 to solve for \( x \). This results in \( x = 6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions in Equations
Fractions can often be found in equations, and they can sometimes make the solution process seem more complicated than it really is. When fractions appear, it's crucial to understand how they affect the equation. Think of fractions like normal numbers, but they are divided into parts. Here's how to handle them easily:
  • A fraction has a top part, called the numerator, and a bottom part, called the denominator.
  • In an equation, fractions work the same way as whole numbers, but they represent a part of a whole.
  • Pay attention to the denominator. It tells you how many parts you are dividing something into.
To solve equations with fractions, your goal is to handle these parts and simplify the equation wherever possible. This often involves the next key concept鈥攆inding a common denominator.
Common Denominators
A common denominator is a shared multiple of the denominators in fractions involved in an equation. It's like finding a common language or base to simplify communication鈥攐r in this case, calculation.
  • Look at all the denominators in the equation.
  • Find the smallest number that all these denominators can divide into without leaving a remainder. This is the least common denominator (LCD).
  • Use the LCD to eliminate the fractions by multiplying every term in the equation by this common denominator.
For example, in the equation \( \frac{2}{9}x - \frac{1}{3} = 1 \), the LCD is 9. Multiplying every term by 9 removes the fractions, turning the equation into \( 2x - 3 = 9 \). Using a common denominator simplifies the fractions, enabling you to solve the equation more straightforwardly.
Isolating Variables
Isolating the variable is a crucial step in solving equations. It means getting the variable all by itself on one side of the equation. This way, you can figure out exactly what value the variable holds.
  • Once fractions are eliminated using a common denominator, focus on moving all other terms away from the variable.
  • Use basic operations like addition, subtraction, multiplication, and division to isolate the variable.
  • If the variable is part of a term, deal with that term separately. For example, if the equation is \( 2x - 3 = 9 \), add 3 to both sides to get \( 2x = 12 \).
Finally, divide by the coefficient of the variable to find its value. So, dividing both sides by 2 gives us \( x = 6 \). Isolating variables step-by-step ensures the solution is correct and simplifies understanding.

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Most popular questions from this chapter

Perform the indicated operations. See Sections 1.5 and 1.6 $$ -5-(-1) $$

Place \(\leq,>,\) or \(=\) in the appropriate space to make each a true statement. See Sections \(1.2,1.4,\) and 1.7 $$ |-5|-(-5) $$

Let \(x=1\) and then \(x=2\) in the equation \(x+5=x+6 .\) Is either number a solution? How many solutions do you think this equation has? Explain your answer.

Explain how solving a linear inequality is similar to solving a compound inequality.

To "break even "in a manufacturing business, revenue \(R\) (income) must equal the cost \(C\) of production, or \(R=C\). The cost \(C\) of producing \(x\) number of paperback books is given by \(C=4.50 x+2400 .\) Income \(R\) from these books is given by \(R=7.50 x .\) Find how many books should be produced and sold to break even.
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