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To perform the operation and combine the two fractions into one, we start by identifying the given fractions:

\[
\frac{7x}{x - 1} + \frac{5}{x + 3}
\]

### Step 1: Find a Common Denominator

The common denominator for the fractions is the product of their individual denominators:

\[
(x - 1)(x + 3)
\]

### Step 2: Rewrite Each Fraction with the Common Denominator

Now, rewrite each fraction with the common denominator:

\[
\frac{7x}{x - 1} = \frac{7x(x + 3)}{(x - 1)(x + 3)}
\]
\[
\frac{5}{x + 3} = \frac{5(x - 1)}{(x - 1)(x + 3)}
\]

### Step 3: Combine the Fractions

Now add the two fractions together:

\[
\frac{7x(x + 3) + 5(x - 1)}{(x - 1)(x + 3)}
\]

### Step 4: Expand and Simplify the Numerator

Expand the terms in the numerator:

\[
7x(x + 3) = 7x^2 + 21x
\]
\[
5(x - 1) = 5x - 5
\]

Combine these:

\[
7x^2 + 21x + 5x - 5 = 7x^2 + 26x - 5
\]

### Step 5: Write the Final Combined Fraction

The final combined fraction is:

\[
\frac{7x^2 + 26x - 5}{(x - 1)(x + 3)}
\]

This is the expression in a single fraction.

Would you like more details, or do you have any questions?

### Related Questions:

1. How do you find the least common denominator for more complex fractions?
2. What are the general rules for adding and subtracting fractions with different denominators?
3. How can we simplify complex rational expressions further?
4. What is the importance of factoring in simplifying fractions?
5. How would the process differ if the fractions had multiple variables?

**Tip:** Always factor the denominators completely before finding a common denominator for rational expressions.

Perform the operation and combine to one fraction: 7x/(x-1) + 5/(x+3)

Learn how to add the algebraic fractions 7x/(x-1) + 5/(x+3) by finding a common denominator and simplifying the expression. This solution demonstrates key concepts in rational expressions and algebraic manipulation, suitable for Grades 9-11.

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