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Chapter 3: Problem 2

Solve. $$\frac{1}{3}-\frac{5}{6}=\frac{1}{x}$$

Short Answer

Expert verified
x = -2

Step by step solution

01

Determine the Common Denominator

Find the common denominator of the fractions on the left-hand side of the equation. The denominators are 3 and 6. The least common multiple of 3 and 6 is 6.
02

Convert Fractions

Convert \( \frac{1}{3} \) to a fraction with 6 as the denominator: \( \frac{1}{3} = \frac{2}{6} \). Now the equation becomes \( \frac{2}{6} - \frac{5}{6} = \frac{1}{x} \).
03

Subtract the Fractions

Subtract the fractions on the left-hand side: \( \frac{2}{6} - \frac{5}{6} = \frac{-3}{6} \). Simplify this fraction to \( \frac{-1}{2} \). Now the equation is \( \frac{-1}{2} = \frac{1}{x} \).
04

Solve for x

Set up the equation \( \frac{-1}{2} = \frac{1}{x} \). Cross-multiply to get \( -x = 2 \). Finally, solve for \( x \) by dividing both sides by -1: \( x = -2 \).

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Key Concepts

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

Understanding the Common Denominator
To subtract fractions, we first need a shared base, called the common denominator. Without a common denominator, the fractions are incompatible for straightforward subtraction. To find the common denominator between two fractions, we look for the least common multiple (LCM) of their denominators. For instance, in the equation \( \frac{1}{3} - \frac{5}{6} = \frac{1}{x} \), the denominators are 3 and 6. The LCM of 3 and 6 is 6. Converting \( \frac{1}{3} \) to a fraction with 6 as the denominator, we get \( \frac{2}{6} \). Now, it鈥檚 easy to work with \( \frac{2}{6} - \frac{5}{6} \).
Fraction Simplification
Simplifying fractions can help make problems easier to solve. After finding a common denominator and performing the subtraction or addition, you may have a fraction that can be simplified further. In our example, after subtracting the fractions \( \frac{2}{6} - \frac{5}{6} \), we get \( \frac{-3}{6} \). By simplifying \( \frac{-3}{6} \), we divide the numerator and the denominator by their greatest common divisor (which is 3). This gives us \( \frac{-1}{2} \). Simplification is crucial as it often leads to a more straightforward solution and clearer final answer.
Cross-Multiplication in Equations
Cross-multiplication is a method used to solve equations involving fractions. It helps us get rid of the fractions so that the equation is easier to work with. In the example \( \frac{-1}{2} = \frac{1}{x} \), we can cross-multiply to solve for \( x \). To cross-multiply, we multiply the numerator of one fraction by the denominator of the other and set them equal to each other: \(-1 \times x = 2 \times 1\). This simplifies to \( -x = 2 \). Solving for \( x \), we divide both sides by -1, resulting in \( x = -2 \). Cross-multiplication is a powerful tool in algebra that simplifies solving fractions and ratios.

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Most popular questions from this chapter

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