- Analyze the given rational expressions to find which one simplifies to \(\frac{3}{2}\).
- Simplify the first expression \(\frac{3y}{2y-6} + \frac{9}{6-2y}\) by rewriting the second denominator as \(-(2y-6)\).
- Combine the terms to get \(\frac{3y-9}{2y-6}\) and factor to identify common terms: \(\frac{3(y-3)}{2(y-3)}\).
- Simplify by canceling \((y-3)\) to obtain \(\boxed{\frac{3y}{2y-6} + \frac{9}{6-2y}}\).
Explanation
- Problem Analysis and Candidate Expressions
We are tasked with determining which of the four given rational expressions simplifies to the constant value \(\frac{3}{2}\). The four candidates are:
- \(\frac{3y}{2y-6} + \frac{9}{6-2y}\)
- \(\frac{9}{y} - \frac{6}{y+2}\)
- \(\frac{3y}{2y-6} + \frac{9}{2y-6}\)
- \(\frac{6}{2y} - \frac{9y}{2-3y}\)
We will simplify each expression one by one to see which one results in \(\frac{3}{2}\).
- Evaluating the First Expression
Let's examine the first expression:
\[ \frac{3y}{2y-6} + \frac{9}{6-2y} \]
Notice that the second denominator \(6-2y\) can be written as \(-(2y-6)\). Thus, the expression becomes:
\[ \frac{3y}{2y-6} + \frac{9}{-(2y-6)} = \frac{3y}{2y-6} - \frac{9}{2y-6} \]
Now, combine the terms over the common denominator \(2y-6\):
\[ \frac{3y-9}{2y-6} \]
Factor both the numerator and the denominator:
\[ \frac{3(y-3)}{2(y-3)} \]
Assuming \(y
eq 3\), we can cancel out the common factor \((y-3)\):
\[ \frac{3}{2} \]
This matches the target value! We have found the correct expression.
- Verification of Other Expressions
For completeness, let's briefly look at the others to ensure they do not simplify to \(\frac{3}{2}\):
- The second expression \(\frac{9}{y} - \frac{6}{y+2}\) simplifies to \(\frac{9(y+2) - 6y}{y(y+2)} = \frac{3y+18}{y^2+2y}\), which clearly depends on \(y\).
- The third expression \(\frac{3y+9}{2y-6} = \frac{3(y+3)}{2(y-3)}\), which also depends on \(y\).
- The fourth expression \(\frac{6}{2y} - \frac{9y}{2-3y} = \frac{3}{y} - \frac{9y}{2-3y} = \frac{3(2-3y)-9y^2}{y(2-3y)} = \frac{6-9y-9y^2}{2y-3y^2}\), which again depends on \(y\).
Only the first expression is equivalent to \(\frac{3}{2}\).
- Final Conclusion
The expression \(\frac{3y}{2y-6} + \frac{9}{6-2y}\) simplifies correctly to the constant \(\frac{3}{2}\). Therefore, this is the correct choice. The final answer is \(\boxed{\frac{3y}{2y-6} + \frac{9}{6-2y}}\).
Examples
In algebra, simplifying rational expressions is a vital skill used to analyze complex systems, such as finding equilibrium points in economic models or solving for variables in physical science formulas. By reducing complicated-looking fractions to a single constant, we can easily see how different components interact and cancel each other out, leading to a much simpler and more useful model for predicting real-world outcomes.