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Chapter 11: Problem 28

Solve the equation. $$\frac{1}{5}-\frac{2}{5 x}=\frac{1}{x}$$

Short Answer

Expert verified
The solution to the equation is \(x = 7\).

Step by step solution

01

Clear the fractions

To get rid of fractions, find the least common multiple (LCM) of the denominators and multiply each term of the equation with it. The LCM of 5, \(5x\), and \(x\) is \(5x\). Then the equation becomes: \(x - 2 = 5\)
02

Simplify and isolate the variable

Simplify the equation by getting the terms with \(x\) on one side and the constants on the other. Doing it, you get \(x = 2 + 5\)
03

Solve for the variable

This gives the final answer as \(x = 7\)

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Key Concepts

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

Fraction Equations
Fraction equations involve variables and numbers in fractional form. Unlike regular equations, they require special attention since you're dealing with numerators and denominators. To solve a fraction equation, you often need to eliminate the fractions. This makes the equation easier to handle.

In the given problem, you have a fraction equation: \(\frac{1}{5}-\frac{2}{5x}=\frac{1}{x}\). Notice each term is a fraction involving a variable or number.

It's important to handle these carefully by doing steps like clearing fractions using the Least Common Multiple (LCM). This helps to make the equation simpler to solve. We'll discuss how to use LCM effectively in the next section.
Least Common Multiple
The Least Common Multiple (LCM) is essential when solving fraction equations. It helps in combining and eliminating fractions.
  • The LCM is the smallest positive number that is a multiple of each denominator.
  • For our equation, the denominators are 5, \(5x\), and \(x\).

To find the LCM of these, consider the highest degree of each factor that appears.
  • The LCM is \(5x\). It includes both 5 and \(x\), the factors from the denominators.

Once you've identified the LCM, multiply every term in the equation by \(5x\). This step clears the fractions, leaving you with a simpler equation to solve: \(x - 2 = 5\).

Using the LCM consolidates the equation and paves the way toward isolating variables.
Isolating Variables
Isolating variables is the final step to solve equations after clearing fractions. It means rearranging the equation to have the variable on one side.

From the previous step, after clearing fractions, our equation is \(x - 2 = 5\). Here, the goal is to get \(x\) by itself on one side.

To isolate \(x\), add 2 to both sides of the equation:
  • Start with: \(x - 2 = 5\).
  • Adding 2 gives us: \(x = 5 + 2\).

After simplification, the result is \(x = 7\).

Isolating the variable allows you to find its value and solve the equation, making it clear and straightforward.

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Most popular questions from this chapter

You will write and simplify a general expression for the average speed traveled when making a round trip. Let \(d\) represent the one-way distance. Let \(x\) represent the speed while traveling there and let \(y\) represent the speed while traveling back. What do you notice about the variables in the final answer? If your distance is doubled what happens to the average speed?

You can use \(x-3\) as the LCD when finding the sum \(\frac{5}{x-3}+\frac{2}{3-x}\) What number can you multiply the numerator and the denominator of the second fraction by to get an equivalent fraction with \(x-3\) as the new denominator?

Simplify the radical expression. $$\sqrt{72}$$

Evaluate the function for \(x=0,1,2,3,\) and 4. $$f(x)=-x+9 \quad$$

Simplify. \(\frac{2 m}{3} \cdot 6 m^{2}\)
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