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A common denominator is like a common language for fractions. When we say fractions need a common denominator for certain operations, it means they need to share the same bottom number. This is essential for adding or subtracting fractions.
If two fractions like \( \frac{2x}{y} \) and \( \frac{x}{y} \) have the same denominator, you are in luck! You don't need to change anything, you can proceed directly to the next step of your operation. This simplifies things a lot, as it allows you to focus on just the numerators.
In cases where denominators are not the same, you'll need to find a least common denominator (LCD). This involves finding the least common multiple (LCM) of the denominators. But in the exercise given, since both fractions already have the same denominator, \( y \), we can skip this step and make subtraction straightforward.

Once you have fractions with a common denominator, you're ready to perform the subtraction. Subtraction of fractions works a lot like a subtraction of plain numbers, with a bit of a twist.
Here’s a simple rundown of the steps to follow:

• Keep the common denominator as it is.
• Subtract the numerators, just as you would with regular numbers.

So, to subtract \( \frac{2x}{y} - \frac{x}{y} \), you simply take the numerators (the top numbers) and subtract one from the other: \( 2x - x \).
This leaves you with \( \frac{x}{y} \). It's that simple! The key is keeping the denominator the same while applying the operation to the numerators.

Simplifying expressions is like tidying up a messy room. You want the expression to be as neat and simple as possible. After subtracting fractions, like in our exercise, you might find your resulting expression can be boiled down even further.
The expression \( \frac{x}{y} \) is already quite simple. But simplification is especially useful in more complex expressions where terms can cancel each other out, or coefficients can be factored out.
Review your expression for any opportunities to combine like terms or use algebraic identities. In our case, after subtraction, \( \frac{x}{y} \) was as simple as it could get. Always have a quick glance to make sure there’s no extra "clutter" that can be removed.