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To solve the given problem, we need to simplify the rational expression:

\[
\frac{1}{y} + \frac{12}{y^2 - 100} - \frac{6}{y^2 - 10y}
\]

### Step 1: Factor the Denominators
We first factor the denominators:

1. \( y^2 - 100 \) is a difference of squares:
   \[
   y^2 - 100 = (y - 10)(y + 10)
   \]

2. \( y^2 - 10y \) has a common factor of \( y \):
   \[
   y^2 - 10y = y(y - 10)
   \]

### Step 2: Find the Least Common Denominator (LCD)
The LCD is the product of all the unique factors from the denominators:
\[
\text{LCD} = y(y - 10)(y + 10)
\]

### Step 3: Rewrite Each Term with the LCD as the Denominator
Adjust each fraction to have the common denominator:

1. For \(\frac{1}{y}\):
   \[
   \frac{1}{y} = \frac{(y - 10)(y + 10)}{y(y - 10)(y + 10)}
   \]

2. For \(\frac{12}{y^2 - 100}\):
   \[
   \frac{12}{(y - 10)(y + 10)} = \frac{12y}{y(y - 10)(y + 10)}
   \]

3. For \(\frac{6}{y^2 - 10y}\):
   \[
   \frac{6}{y(y - 10)} = \frac{6(y + 10)}{y(y - 10)(y + 10)}
   \]

### Step 4: Combine the Fractions
Now, we can combine the fractions over the common denominator:
\[
\frac{(y - 10)(y + 10) + 12y - 6(y + 10)}{y(y - 10)(y + 10)}
\]

### Step 5: Simplify the Numerator
Expand and combine like terms in the numerator:

1. Expand \((y - 10)(y + 10)\):
   \[
   (y - 10)(y + 10) = y^2 - 100
   \]

2. Expand and simplify the remaining terms:
   \[
   y^2 - 100 + 12y - 6(y + 10) = y^2 - 100 + 12y - 6y - 60 = y^2 + 6y - 160
   \]

### Step 6: Write the Final Simplified Expression
The simplified expression is:
\[
\frac{y^2 + 6y - 160}{y(y - 10)(y + 10)}
\]

This is the simplified form of the given rational expression.

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

1. How do you find the least common denominator (LCD) for multiple fractions?
2. How do you simplify a rational expression after combining fractions?
3. How do you factor quadratic expressions like \( y^2 - 100 \)?
4. What are common mistakes to avoid when adding or subtracting rational expressions?
5. How do you handle subtraction with rational expressions?

**Tip:** When working with rational expressions, always factor completely before finding the least common denominator (LCD).

Learn how to simplify the rational expression 1/y + 12/(y^2 - 100) - 6/(y^2 - 10y) using factoring and finding the least common denominator (LCD). Suitable for students in Grades 9-12, this problem demonstrates factoring techniques and simplifying rational expressions.

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