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.