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Chapter 6: Problem 31

Solve the given equations and check the results. $$\frac{1}{x^{2}-x}-\frac{1}{x}+\frac{1}{1-x}=0$$

Short Answer

Expert verified
There are no solutions; x = 1 is invalid due to division by zero.

Step by step solution

01

Simplify the Equation

Start by combining the fractions on the left-hand side. The expression is \( \frac{1}{x^{2}-x} - \frac{1}{x} + \frac{1}{1-x} \). To combine these, find a common denominator. Notice that \( x^{2}-x = x(x-1) \) and \( 1-x = -(x-1) \). Therefore, the common denominator is \( x(x-1)(1-x) \).
02

Rewrite Each Term

Rewrite each term with the common denominator: - \( \frac{1}{x^{2}-x} = \frac{1}{x(x-1)} = \frac{1-x}{x(x-1)(1-x)} \)- \( \frac{1}{x} = \frac{(1-x)^{2}}{x(x-1)(1-x)} \)- \( \frac{1}{1-x} = \frac{x}{x(x-1)(1-x)} \)
03

Combine the Fractions

Combine the fractions over the common denominator: \[ \frac{1-x}{x(x-1)(1-x)} - \frac{(1-x)^{2}}{x(x-1)(1-x)} + \frac{x}{x(x-1)(1-x)} = \frac{-x^{2} + 2x - 1}{x(x-1)(1-x)} \].
04

Solve the Numerator

Since the overall equation equals zero, we focus on the numerator: Solve \(-x^{2} + 2x - 1 = 0\), by factoring or using the quadratic formula: Using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where \( a = -1, b = 2, c = -1\).Compute the discriminant: \( 2^2 - 4(-1)(-1) = 4 - 4 = 0 \). Therefore, \( x = \frac{-2 \pm 0}{-2} = 1 \).
05

Verify the Solution

Check for potential restrictions and verify solution if it doesn鈥檛 make any denominator zero. The solution \( x = 1 \) makes the term \( \frac{1}{x^2-x} \) undefined because it leads to division by zero. Thus, there are no viable solutions.

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Key Concepts

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

Common Denominator
When dealing with rational equations such as \( \frac{1}{x^2-x} - \frac{1}{x} + \frac{1}{1-x} = 0 \), finding a common denominator is essential.This means determining a shared base for all fractions involved.
In the given equation, each denominator is unique: \( x^2-x \), \( x \), and \( 1-x \).
We'll transform each expression to have the same denominator, simplifying the process to add or subtract the fractions.
  • First, notice that \( x^2-x \) can be factored into \( x(x-1) \).
  • Also, recognize \( 1-x \) can be written as \( -(x-1) \).
  • Thus, the common base becomes \( x(x-1)(1-x) \), as it incorporates all individual factors from the denominators.
This shared denominator allows us to rewrite and combine the fractions into one expression. This is a powerful tool that makes solving these types of equations more manageable.
Quadratic Formula
The quadratic formula is a fundamental method to solve equations of the form \( ax^2 + bx + c = 0 \).
Our equation simplifies to solving the quadratic \(-x^2 + 2x - 1 = 0\).
Using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]we identify the coefficients from our equation:
  • \(a = -1\)
  • \(b = 2\)
  • \(c = -1\)
We calculate the discriminant \(b^2 - 4ac\):
  • \(2^2 - 4(-1)(-1) = 4 - 4 = 0\).
Since the discriminant is zero, there's exactly one real root.
The solution is \(x = \frac{-2}{-2} = 1\). This critical step is often employed in algebra when factoring is complex or impossible.
Fraction Simplification
Simplifying fractions is a crucial skill for solving rational equations.It lets us reduce expressions to their most basic form, which is necessary for addition or subtraction.
For example, let's look at our fractions with a common denominator:
  • The individual terms become \(\frac{1-x}{x(x-1)(1-x)}\), \(\frac{(1-x)^2}{x(x-1)(1-x)}\), and \(\frac{x}{x(x-1)(1-x)}\).
Each fraction is represented with the same base, allowing direct combination:
\[\frac{1-x - (1-x)^2 + x}{x(x-1)(1-x)} = \frac{-x^2 + 2x - 1}{x(x-1)(1-x)}.\]By combining them, we focus on the numerator, as the denominator is consistent.Through simplification, handling fractions becomes straightforward.
This reduces errors and provides clarity, making it a vital process in algebraic manipulation.

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Most popular questions from this chapter

After finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{x^{2}-x-2}{x^{2}-x}\) (b) \(\frac{x^{2}-x-2}{x^{2}+x}\)

Set up appropriate equations and solve the given stated problems. All numbers are accurate to at least two significant digits. One company determines that it will take its crew \(450 \mathrm{h}\) to clean up a chemical dump site, and a second company determines that it will take its crew \(600 \mathrm{h}\) to clean up the site. How long will it take the two crews working together?

Solve for the indicated letter. Each of the given formulas arises in the technical or scientific area of study listed. $$\frac{2}{3}=\frac{h}{x}+\frac{1}{6 x}, \text { for } x$$

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{1}{R^{2}}+\left(\omega C-\frac{1}{\omega L}\right)^{2} \text { (electricity) }$$

Solve the given equations and check the results. $$\frac{1}{2 x+3}=\frac{5}{2 x}-\frac{4}{2 x^{2}+3 x}$$
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