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When dealing with fractional equations, finding a common denominator is a crucial step that allows us to combine fractions easily. A common denominator is a shared multiple of the denominators of two or more fractions. In other words, it's a number that can be evenly divided by each of the original denominators.

For instance, in the exercise \( \frac{x}{10} + \frac{x}{5} = 20 \), the denominators are 10 and 5. The smallest number that both 10 and 5 divide evenly into is 10, which is thus chosen as the common denominator.

Using a common denominator allows you to rewrite the fractions so they have the same denominator, like so:

• \( \frac{x}{10} \) remains \( \frac{x}{10} \)
• \( \frac{x}{5} \) is converted to \( \frac{2x}{10} \)

Finding a common denominator simplifies the addition or subtraction of fractions since you only need to combine the numerators once the denominators match.

Once you have a common denominator, the next step is to simplify the equation by combining like terms. Simplifying means rewriting the equation in a form that's easier to work with, reducing complexity without changing the equation's basic properties.

In the exercise, after finding the common denominator and rewriting the fractions as \( \frac{x}{10} + \frac{2x}{10} \), we can combine the numerators since the denominators are now the same:

• Combine \( x \) and \( 2x \) to get \( 3x \).

This gives us the new simplified form of the equation: \( \frac{3x}{10} = 20 \). Simplifying the equation makes it easier to manage in further processing steps, as it reduces the number of fractional terms involved.

The final stage in handling fractional equations is solving for the variable. This typically involves isolating the variable on one side of the equation.

For our exercise, the equation was simplified to \( \frac{3x}{10} = 20 \). To solve for \( x \), we undertake some straightforward algebraic manipulations:

• First, eliminate the fraction by multiplying both sides of the equation by 10, giving \( 3x = 200 \).
• Next, solve for \( x \) by dividing both sides by 3, leading to \( x = \frac{200}{3} \).

Through these steps, the variable \( x \) is isolated and its value determined. Solving for variables is a fundamental concept in algebra, as it allows us to find unknown quantities in equations.