Short Answer 
The expression (frac{frac{5}{a -3} – 4}{2 + frac{1}{a – 3}}) is simplified by rewriting both the numerator and denominator with a common denominator of (a – 3), leading to the form (frac{17 – 4a}{2a – 5}). This final result represents a more manageable version of the original expression, confirming the effectiveness of the simplification process.
Step 1: Setup the Expression
Start by identifying the given expression, which is (frac{frac{5}{a -3} – 4}{ 2 + frac{1}{a – 3}}). You will need to express both the numerator and the denominator in terms of a common denominator to simplify it effectively. This involves rewriting the terms such that they both share (a – 3) as their denominator.
- Rewrite the numerator: (frac{5}{a – 3} – 4 = frac{5 – 4a + 12}{a – 3})
- Rewrite the denominator: (2 + frac{1}{a – 3} = frac{2a – 6 + 1}{a – 3})
Step 2: Cancel Denominators
Now that you have a common denominator, the next step is to cancel the denominators from both the numerator and the denominator. This simplifies the expression considerably and allows you to focus only on the numerators which you will now divide.
- By dividing the numerators, you get: (frac{5 – 4a + 12}{2a – 6 + 1})
- Combine like terms: the numerator simplifies to 17 – 4a and the denominator simplifies to 2a – 5.
Step 3: Final Simplified Expression
After simplifying further, your expression can be expressed as (frac{17 – 4a}{2a – 5}). This final result showcases the simplified form of the original expression, making it easy to work with in future calculations or proofs.
- Final Result: (frac{17 – 4a}{2a – 5})
- Conclusion: The method used confirms the effective handling of fractions by recognizing shared denominators and simplification techniques.