91Ó°ÊÓ

When simplifying algebraic fractions, the first step is finding a common denominator. A common denominator allows us to have a uniform base for both fractions, making it easier to combine them later. For the given fractions \(\frac{t}{t-5}\) and \(\frac{t-1}{t+5}\), their denominators are \(t-5\) and \(t+5\), respectively. To find a common ground, multiply the two denominators together: \((t-5)(t+5)\). This creates a shared denominator and sets the stage for the next step of rewriting each fraction with this common base. \
Always aim to use the least common multiple (LCM) of the denominators if feasible. This simplifies later simplification steps. Here, our LCM is efficiently \((t-5)(t+5)\).

Once you have the common denominator, the next task is rewriting the fractions to share it. We achieve this by adjusting the numerators to reflect this change. In our example, we rewrite \(\frac{t}{t-5}\) and \(\frac{t-1}{t+5}\) to have \((t-5)(t+5)\) as the denominator. \
To do this: \

• Multiply both the numerator and denominator of \(\frac{t}{t-5}\) by \(t+5\), resulting in \(\frac{t(t+5)}{(t-5)(t+5)}\).
• Similarly, multiply both the numerator and denominator of \(\frac{t-1}{t+5}\) by \(t-5\), producing \(\frac{(t-1)(t-5)}{(t-5)(t+5)}\).

Now, expand each numerator: \(\frac{t(t+5)}{(t-5)(t+5)} = \frac{t^2 + 5t}{(t-5)(t+5)}\), and \(\frac{(t-1)(t-5)}{(t-5)(t+5)} = \frac{t^2 - 6t + 5}{(t-5)(t+5)}\).
Expanding numerators makes it easier to combine the fractions in the next step.

After rewriting the fractions with a common denominator and expanding the numerators, you can combine them into one. Remember, both fractions now have the same denominator, so we simply combine the numerators over this denominator. \
For this exercise: \

• Write \(\frac{t^2 + 5t}{(t-5)(t+5)} - \frac{t^2 - 6t + 5}{(t-5)(t+5)}\).
• Since the denominators are the same: \(\frac{(t^2 + 5t) - (t^2 - 6t + 5)}{(t-5)(t+5)}\).

When combining fractions, ensure to distribute the negative sign correctly. Combining fractions accurately paves the way for final simplification.

The last step is to simplify the expression, focusing on the numerator. Simplifying helps to condense the expression to its simplest form. Continuing from our combined fractions: \(\frac{(t^2 + 5t) - (t^2 - 6t + 5)}{(t-5)(t+5)}\). \
Follow these sub-steps: \

• Distribute the negative sign to the terms inside the parentheses: \(t^2 + 5t - t^2 + 6t - 5\).
• Combine like terms: \(\frac{11t - 5}{(t-5)(t+5)}\).

Ensure that each step is performed accurately to avoid errors. This final algebraic fraction is now in its simplest form. \
Remember, simplification means making the expression as neat and manageable as possible, but also cross-checking to guarantee no terms were missed or incorrectly combined.