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When dealing with algebraic equations involving fractions, finding a common denominator is a crucial skill. It allows you to seamlessly add or subtract fractions by ensuring they share the same denominator. In the equation \[ \frac{x}{6} - \frac{4x}{3} = \frac{1}{9} \]the fractions on the left have different denominators of 6 and 3. In such cases, identifying the least common denominator (LCD) simplifies the process. For 6 and 3, the LCD is 6. This is because 6 is the smallest number that both 6 and 3 divide into evenly. Once a common denominator is determined, fractions can be adjusted so they facilitate operations without changing the equation’s intrinsic value. To convert a fraction to an equivalent one with the common denominator, multiply both the numerator and denominator by the necessary factor. For example, to convert \( \frac{4x}{3} \) to a denominator of 6, multiply by 2 to get \( \frac{8x}{6} \). This step places both fractions over a common denominator, allowing them to be easily combined.

Operating with fractions often involves addition, subtraction, multiplication, or division. For direct addition or subtraction, fractions must first have a common denominator. This means rearranging each fraction so they share the same bottom number, which simplifies combining them. Let's look at our problem again: \[ \frac{x}{6} - \frac{8x}{6} = \frac{1}{9} \]. Once the fractions have been rewritten with the same denominator, you can perform addition or subtraction straightforwardly:

• Keep the common denominator.
• Add or subtract the numerators as indicated.

In the example, subtract the numerators, "\( x - 8x \)," to yield \( \frac{-7x}{6} \). The importance lies in simplifying fractions to easy-to-combine components while ensuring the mathematical balance of the equation is preserved. Only when fractions are united over a common denominator, performing arithmetic operations becomes clear and simple.

Cross multiplication is a useful technique when solving equations with fractions on both sides, especially if they are proportional. It’s an effective way to eliminate fractions by creating a simpler equation to solve. Consider the resulting equation from our exercise: \[ \frac{-7x}{6} = \frac{1}{9} \].In cross multiplication:

• Multiply the numerator of one fraction by the denominator of the other fraction, and vice versa.
• The products form a new equation without fractions, making it easier to solve for the variable — in this case, "\(-63x = 6\)."

Cross multiplication not only clears fractions but also ensures that both sides of the equation remain balanced, as identical cross-multiplying values are preserved. The final step involves isolating the variable by standard algebraic means, like division, which leads to an answer. This technique extends well beyond simple equations and is integral in understanding relationships among ratios and proportions.