To solve the equation \(\frac{x - 1}{x - 5} = \frac{5}{3}\), we can cross-multiply to eliminate the fractions.

### Step 1: Cross-multiply
\[
3(x - 1) = 5(x - 5)
\]

### Step 2: Distribute on both sides
\[
3x - 3 = 5x - 25
\]

### Step 3: Move all the terms involving \(x\) to one side and constant terms to the other side
\[
3x - 5x = -25 + 3
\]

\[
-2x = -22
\]

### Step 4: Solve for \(x\)
\[
x = \frac{-22}{-2} = 11
\]

### Solution:
\[
x = 11
\]

Would you like more details or have any questions? Here are some related questions:

1. How can I check if the solution is correct?
2. What if the equation had a different form, like \(\frac{x - 1}{x + 5} = \frac{5}{3}\)?
3. What are other methods to solve this kind of rational equation?
4. Can you explain how to solve quadratic equations that might result from cross-multiplication?
5. How do we handle situations where cross-multiplying results in no solutions?

**Tip:** Always substitute your solution back into the original equation to verify its correctness.

Learn how to solve the rational equation (x - 1)/(x - 5) = 5/3 with a detailed step-by-step solution. This problem involves cross-multiplication and solving linear equations. Suitable for high school students in Grades 9-12.

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