Reduce the complex fraction ﻿fraction numerator 5 y plus begin display style fraction numerator 1 over denominator x minus 5 end fraction end style over denominator begin display style fraction numerator 3 x plus 1 over denominator 3 end fraction end style end fraction﻿.

To simplify the complex fraction 

\[
\frac{5y + \frac{1}{x - 5}}{\frac{3x + 1}{3}},
\]

we will handle the numerator and the denominator separately before combining them.

### Step 1: Simplify the Denominator

The denominator is \(\frac{3x + 1}{3}\).

To simplify, we'll multiply both the numerator and the denominator by 3:

\[
\frac{3x + 1}{3} = 3x + 1.
\]

### Step 2: Simplify the Numerator

The numerator is \(5y + \frac{1}{x - 5}\).

To combine the terms, let's express everything over a common denominator. The common denominator for \(5y\) and \(\frac{1}{x - 5}\) would be \(x - 5\). Therefore, rewrite \(5y\) with this common denominator:

\[
5y = \frac{5y(x - 5)}{x - 5} = \frac{5yx - 25y}{x - 5}.
\]

Thus, the numerator becomes:

\[
5y + \frac{1}{x - 5} = \frac{5yx - 25y + 1}{x - 5}.
\]

### Step 3: Form the Complex Fraction

Now, rewrite the entire complex fraction using the simplified numerator and denominator:

\[
\frac{\frac{5yx - 25y + 1}{x - 5}}{3x + 1}.
\]

### Step 4: Simplify the Complex Fraction

To simplify the complex fraction, divide the numerator by the denominator. This can be done by multiplying by the reciprocal of the denominator:

\[
\frac{5yx - 25y + 1}{x - 5} \times \frac{1}{3x + 1} = \frac{5yx - 25y + 1}{(x - 5)(3x + 1)}.
\]

### Final Simplified Form

The simplified form of the complex fraction is:

\[
\frac{5yx - 25y + 1}{(x - 5)(3x + 1)}.
\]

This is the reduced form of the given complex fraction.

Would you like to go over any step in more detail, or do you have any other questions?

Here are some additional questions related to this topic:

1. How do you find a common denominator for adding fractions?
2. What are the steps for multiplying and dividing fractions?
3. How do you simplify complex fractions with multiple terms in both the numerator and the denominator?
4. What are some methods for checking if a fraction is fully simplified?
5. How do you factor polynomials to simplify algebraic expressions?

**Tip:** When simplifying complex fractions, always start by simplifying the numerator and the denominator separately before combining them.

Learn how to simplify the complex fraction with numerator 5y + 1/(x - 5) step-by-step. This problem covers complex fractions and fraction simplification concepts, suitable for advanced high school students.

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