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The given equation is:

\[
\frac{3}{x - 6} = \frac{1}{x - 1} + \frac{7}{(x - 1)(x - 6)}
\]

To solve this equation, we'll first eliminate the denominators by multiplying through by the common denominator, which is \((x - 6)(x - 1)\).

### Step-by-step process:

1. Multiply both sides by \((x - 6)(x - 1)\) to eliminate the fractions:
   \[
   (x - 6)(x - 1) \cdot \frac{3}{x - 6} = (x - 6)(x - 1) \cdot \frac{1}{x - 1} + (x - 6)(x - 1) \cdot \frac{7}{(x - 1)(x - 6)}
   \]

2. Simplify each term:
   \[
   3(x - 1) = (x - 6) + 7
   \]

3. Expand both sides:
   \[
   3x - 3 = x - 6 + 7
   \]

4. Simplify the right-hand side:
   \[
   3x - 3 = x + 1
   \]

5. Move all terms involving \(x\) to one side:
   \[
   3x - x = 1 + 3
   \]

6. Simplify:
   \[
   2x = 4
   \]

7. Solve for \(x\):
   \[
   x = 2
   \]

### Solution:
The solution to the equation is \(x = 2\).

Would you like further details on any steps? Here are some related questions:

1. How do you determine the least common denominator for rational expressions?
2. Can you explain more about how to expand and simplify algebraic expressions?
3. How would the equation change if different constants were used in the numerators?
4. What are the restrictions on the values of \(x\) for this equation?
5. How does multiplying both sides by the common denominator help solve rational equations?

**Tip:** Always check for extraneous solutions by substituting your answer back into the original equation. For this equation, \(x = 6\) or \(x = 1\) would cause division by zero, so they must be excluded.

Solve the equation: \( \frac{3}{x - 6} = \frac{1}{x - 1} + \frac{7}{(x - 1)(x - 6)} \)

Learn how to solve the rational equation \( \frac{3}{x - 6} = \frac{1}{x - 1} + \frac{7}{(x - 1)(x - 6)} \) using a step-by-step approach. This problem covers key algebraic concepts, such as rational expressions and multiplying both sides by the least common denominator. Suitable for students in Grades 9-11.

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