Why do we have to express it with the same denominator and not cancel since both are  3(x-5)/x+5

It sounds like you're referring to a situation involving fractions where you are dealing with the expression \( \frac{3(x-5)}{x+5} \). When handling such expressions, it's important to carefully follow algebraic rules, especially when combining fractions or simplifying expressions.

Here's why we express fractions with the same denominator instead of just canceling them:

### 1. **Understanding Cancellation**:
   You can only cancel terms in a fraction if the terms are factors of both the numerator and denominator. For example, if both the numerator and denominator have a common factor, you can simplify the expression by canceling that factor. However, **terms that are being added or subtracted cannot be canceled directly**.

   In the case of \( \frac{3(x-5)}{x+5} \), you can't cancel \( x-5 \) and \( x+5 \), because they are not factors of each other; they are distinct expressions due to the difference in signs. The expression \( x-5 \) and \( x+5 \) are not the same and don't simplify through cancellation.

### 2. **Expressing with the Same Denominator**:
   If you are adding or subtracting fractions, you must express them with the same denominator. For example, if you had:
   \[
   \frac{3(x-5)}{x+5} + \frac{2}{x+5}
   \]
   you would already have a common denominator of \( x+5 \), and thus you can combine the numerators:
   \[
   \frac{3(x-5) + 2}{x+5}
   \]
   This simplifies the expression in a proper algebraic manner.

If you're working with an expression that looks similar, you'd need to follow these steps rather than cancel terms unless they are factors common to both the numerator and denominator.

Would you like more details or clarification on this process?

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Here are 5 questions to consider:
1. How do you identify common factors between the numerator and denominator in an expression?
2. What is the difference between simplifying by canceling and finding a common denominator?
3. How do you handle terms that include addition or subtraction when simplifying fractions?
4. Can you always simplify rational expressions by canceling?
5. What techniques help ensure that algebraic expressions are simplified correctly?

**Tip:** Always factor expressions completely before attempting to cancel terms. This avoids mistakes!

Understand why it is necessary to express fractions with the same denominator instead of canceling terms in algebraic expressions like 3(x-5)/(x+5). This explanation covers key concepts in rational expressions and fraction simplification. Suitable for high school students in Grades 9-12.

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