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Chapter 6: Problem 18

Solve. If no solution exists, state this. $$ \frac{2}{3}-\frac{1}{y}=\frac{5}{6} $$

Short Answer

Expert verified
The solution is \(y = -6\).

Step by step solution

01

Isolate the variable term

To solve the equation \(\frac{2}{3}-\frac{1}{y}=\frac{5}{6}\), start by isolating the term with the variable. Subtract \(\frac{2}{3}\) from both sides to get: \(-\frac{1}{y} = \frac{5}{6} - \frac{2}{3}\).
02

Simplify the right-hand side

Find a common denominator to combine \(\frac{5}{6}\) and \(\frac{2}{3}\). The common denominator for 6 and 3 is 6. Convert \(\frac{2}{3}\) to \(\frac{4}{6}\). Now, the equation becomes: \(-\frac{1}{y} = \frac{5}{6} - \frac{4}{6} = \frac{1}{6}\).
03

Solve for y

To isolate \(\frac{1}{y}\), multiply both sides by -1: \(\frac{1}{y} = -\frac{1}{6}\). Then take the reciprocal of both sides to solve for \(y\): \(y = -6\).

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Key Concepts

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

Isolating the variable
In solving rational equations, the first step is often to isolate the term containing the variable. This simplifies the equation and makes it easier to solve. Take the equation: \(\frac{2}{3} - \frac{1}{y} = \frac{5}{6}\). To isolate the variable term \(-\frac{1}{y}\), subtract \(\frac{2}{3}\) from both sides. This gives us: \(-\frac{1}{y} = \frac{5}{6} - \frac{2}{3}\). By isolating the variable, we prepare the equation for further simplification. This method is crucial because it lets us focus on solving one part of the equation at a time.
Common denominator
When dealing with fractions, finding a common denominator is often necessary. This step allows fractions to be added or subtracted easily. In the equation \(\frac{5}{6} - \frac{2}{3}\), the fractions have different denominators. We need to convert \(\frac{2}{3}\) into an equivalent fraction with the same denominator as \(\frac{5}{6}\). The least common denominator (LCD) of 6 and 3 is 6. Hence, \(\frac{2}{3}\) can be written as \(\frac{4}{6}\). Now, we can subtract the fractions: \(\frac{5}{6} - \frac{4}{6} = \frac{1}{6}\). This simplification is essential to keep the equation manageable and solvable.
Reciprocal
A reciprocal of a number is what you multiply that number by to get 1. For example, the reciprocal of \(2\) is \(\frac{1}{2}\). In rational equations, taking reciprocals can help solve for the variable. After isolating the variable term in our equation, we get: \(-\frac{1}{y} = \frac{1}{6}\). To solve for \(y\), multiply both sides by \(-1\): \(\frac{1}{y} = -\frac{1}{6}\). Next, take the reciprocal of both sides: \(y = -6\). Using reciprocals is a powerful technique for solving equations, particularly those involving fractions. This final step reveals the value of the variable and completes the solution.

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Most popular questions from this chapter

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