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Chapter 6: Problem 34

Solve. If no solution exists, state this. $$ -\frac{5}{6}=\frac{1}{x} $$

Short Answer

Expert verified
x = -\frac{6}{5}

Step by step solution

01

Understand the Equation

The equation given is \(-\frac{5}{6} = \frac{1}{x}\). This equation states that the fraction \(-\frac{5}{6}\) is equal to the reciprocal of some number \(x\).
02

Take the Reciprocal

To find \(x\), take the reciprocal of both sides of the equation. The reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\), so we set \(x\) equal to this value.
03

Write the Solution

Since the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\), we find that \(x = -\frac{6}{5}\). There are no restrictions on \(x\) in this context.

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Key Concepts

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

Reciprocal
To solve the exercise, you need to understand the concept of a reciprocal. A reciprocal of a number is what you get when you divide 1 by that number. For a fraction, such as \(-\frac{5}{6}\), its reciprocal is \(-\frac{6}{5}\). Essentially, you flip the numerator and the denominator of the fraction. This idea helps transform the equation into a more straightforward form.

In our problem, \(-\frac{5}{6}\) equals the reciprocal of \(x\), or \(\frac{1}{x}\). By taking the reciprocal of \(-\frac{5}{6}\), we find what \({x}\) should be: \(-\frac{6}{5}\).
Fractions
Fractions are another important concept for solving this problem. A fraction consists of a numerator (the top part) and a denominator (the bottom part). For example, in the fraction \-\frac{5}{6}\, \(-5\) is the numerator and \6\ is the denominator.

When finding the reciprocal of a fraction, just switch the positions of the numerator and the denominator, keeping the sign in mind. So, the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\).

In this exercise, knowing how to handle fractions helps us easily identify \({x}\).
Solving Linear Equations
This exercise also deals with solving linear equations, which are equations of the form \ax + b = 0\. Here we have \(-\frac{5}{6} = \frac{1}{x}\). Solving for \({x}\) involves isolating \({x}\) on one side of the equation.

This linear equation requires taking the reciprocal of both sides first. When you do this, the equation simplifies to \(x = -\frac{6}{5}\), making it easy to identify the value of \({x}\).

Linear equations often involve steps like adding, subtracting, multiplying, or dividing both sides by the same number. Here, taking the reciprocal helped us quickly reach a solution.

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Most popular questions from this chapter

The formula $$a=\frac{\frac{d_{4}-d_{3}}{t_{4}-t_{3}}-\frac{d_{2}-d_{1}}{t_{2}-t_{1}}}{t_{4}-t_{2}}$$ can be used to approximate average acceleration, where the \(d^{\prime}\) 's are distances and the \(t^{\prime}\) s are the corresponding times. Solve for \(t_{1}\).

For each pair of functions fand \(g\), find all values of a for which \(f(a)=g(a)\) $$ f(x)=\frac{1}{1+x}+\frac{x}{1-x}, g(x)=\frac{1}{1-x}-\frac{x}{1+x} $$

Simplify. $$ \frac{u^{6}+v^{6}+2 u^{3} v^{3}}{u^{3}-v^{3}+u^{2} v-u v^{2}} $$

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Simplify. Do not use negative exponents in the answer. $$ \left(-5 x^{2} y^{-6}\right)^{-3} $$
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