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The given expression is:

\[
\frac{x - z^2}{xy} + \frac{2y - x}{y^2}
\]

Let's simplify this step by step.

### Step 1: Find the common denominator
- The denominators are \(xy\) and \(y^2\).
- The least common denominator (LCD) between \(xy\) and \(y^2\) is \(xy^2\).

### Step 2: Rewrite each term with the common denominator
- For the first term: 
\[
\frac{x - z^2}{xy} = \frac{(x - z^2) \cdot y}{xy^2} = \frac{y(x - z^2)}{xy^2}
\]
- For the second term:
\[
\frac{2y - x}{y^2} = \frac{(2y - x) \cdot x}{xy^2} = \frac{x(2y - x)}{xy^2}
\]

### Step 3: Combine the terms
Now that both terms have the common denominator \(xy^2\), we can combine them:
\[
\frac{y(x - z^2)}{xy^2} + \frac{x(2y - x)}{xy^2} = \frac{y(x - z^2) + x(2y - x)}{xy^2}
\]

### Step 4: Expand and simplify the numerator
- Expand the first part:
\[
y(x - z^2) = yx - yz^2
\]
- Expand the second part:
\[
x(2y - x) = 2xy - x^2
\]
So the numerator becomes:
\[
yx - yz^2 + 2xy - x^2 = 3xy - yz^2 - x^2
\]

### Final simplified expression:
\[
\frac{3xy - yz^2 - x^2}{xy^2}
\]

Would you like more details on any of these steps, or have any further questions?

Here are 5 questions related to this:
1. What other methods can be used to simplify rational expressions?
2. How can we find the least common denominator for more complex denominators?
3. Can you provide an example where this method is applied to a different expression?
4. How can factoring help in simplifying rational expressions?
5. What happens if we add rational expressions with variables that include different powers?

**Tip:** When adding rational expressions, always start by finding the least common denominator to combine the terms efficiently.

Simplify the expression: \( \frac{x - z^2}{xy} + \frac{2y - x}{y^2} \)

Learn how to simplify the expression \( \frac{x - z^2}{xy} + \frac{2y - x}{y^2} \) using the least common denominator (LCD) method. This problem demonstrates key concepts in rational expressions and algebra, suitable for students in Grades 9-12.

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