Simplifying algebraic fractions with variables x and y.

Let’s solve each problem one by one. We’re adding or subtracting fractions with variables in the denominators. The key is to find a common denominator, rewrite each fraction with that denominator, then combine the numerators.

---

Problem 1:
\[
\frac{6}{2xy} + \frac{2}{x}
\]

Step 1: Simplify first fraction if possible.
\[
\frac{6}{2xy} = \frac{3}{xy}
\]

Now we have:
\[
\frac{3}{xy} + \frac{2}{x}
\]

Step 2: Common denominator is \( xy \). Rewrite second fraction:
\[
\frac{2}{x} = \frac{2y}{xy}
\]

Step 3: Add numerators:
\[
\frac{3 + 2y}{xy} = \frac{2y + 3}{xy}
\]

✔ Matches the red answer given.

---

Problem 2:
\[
\frac{6}{2xy} + \frac{2}{3x}
\]

Step 1: Simplify first fraction:
\[
\frac{6}{2xy} = \frac{3}{xy}
\]

Now:
\[
\frac{3}{xy} + \frac{2}{3x}
\]

Step 2: Common denominator of \( xy \) and \( 3x \) is \( 3xy \).

Rewrite both:

- \( \frac{3}{xy} = \frac{9}{3xy} \)
- \( \frac{2}{3x} = \frac{2y}{3xy} \)

Step 3: Add:
\[
\frac{9 + 2y}{3xy} = \frac{2y + 9}{3xy}
\]

✔ Matches the red answer.

---

Problem 3:
\[
\frac{6}{2xy} - \frac{2}{3x^2}
\]

Step 1: Simplify first fraction:
\[
\frac{6}{2xy} = \frac{3}{xy}
\]

Now:
\[
\frac{3}{xy} - \frac{2}{3x^2}
\]

Step 2: Common denominator of \( xy \) and \( 3x^2 \) is \( 3x^2y \).

Rewrite:

- \( \frac{3}{xy} = \frac{9x}{3x^2y} \)
- \( \frac{2}{3x^2} = \frac{2y}{3x^2y} \)

Step 3: Subtract:
\[
\frac{9x - 2y}{3x^2y}
\]

✔ Matches the red answer.

---

Problem 4:
\[
\frac{6y}{2xy} + \frac{2}{3x^2}
\]

Step 1: Simplify first fraction:
\[
\frac{6y}{2xy} = \frac{3}{x}
\]

Now:
\[
\frac{3}{x} + \frac{2}{3x^2}
\]

Step 2: Common denominator is \( 3x^2 \).

Rewrite:

- \( \frac{3}{x} = \frac{9x}{3x^2} \)
- \( \frac{2}{3x^2} \) stays same.

Step 3: Add:
\[
\frac{9x + 2}{3x^2} = \frac{2 + 9x}{3x^2}
\]

✔ Matches the red answer.

---

Problem 5:
\[
\frac{y}{2xy} - \frac{2}{3x^3}
\]

Step 1: Simplify first fraction:
\[
\frac{y}{2xy} = \frac{1}{2x}
\]

Now:
\[
\frac{1}{2x} - \frac{2}{3x^3}
\]

Step 2: Common denominator of \( 2x \) and \( 3x^3 \) is \( 6x^3 \).

Rewrite:

- \( \frac{1}{2x} = \frac{3x^2}{6x^3} \)
- \( \frac{2}{3x^3} = \frac{4}{6x^3} \)

Step 3: Subtract:
\[
\frac{3x^2 - 4}{6x^3}
\]

✔ Matches the red answer.

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All problems are solved correctly as shown in the image. Since you didn’t ask for explanation, teaching, or practice — just solving — here’s the final confirmation:

Final Answer:
All five expressions simplify correctly to the red answers shown:
1. $\frac{2y + 3}{xy}$
2. $\frac{2y + 9}{3xy}$
3. $\frac{9x - 2y}{3x^2y}$
4. $\frac{2 + 9x}{3x^2}$
5. $\frac{3x^2 - 4}{6x^3}$