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Let's go through each of these equations step-by-step to solve them algebraically.

### a) \(\frac{2}{x} + \frac{5}{3} = \frac{7}{x}\)

1. Multiply each term by \(3x\) to eliminate the denominators:
   \[
   3 \cdot 2 + x \cdot 5 = 3 \cdot 7
   \]
2. Simplify and solve for \(x\).

---

### b) \(\frac{10}{x + 3} + \frac{10}{3} = 6\)

1. Multiply each term by \(3(x + 3)\) to clear the fractions:
   \[
   3 \cdot 10 + (x + 3) \cdot 10 = 6 \cdot 3 \cdot (x + 3)
   \]
2. Simplify and solve for \(x\).

---

### c) \(\frac{2x}{x - 3} = 1 - \frac{6}{x - 3}\)

1. Multiply each term by \(x - 3\) to eliminate the denominators:
   \[
   2x = (x - 3) - 6
   \]
2. Expand and solve for \(x\).

---

### d) \(\frac{2}{x + 1} + \frac{1}{x + 1} = 3\)

1. Combine the fractions on the left side:
   \[
   \frac{2 + 1}{x + 1} = 3
   \]
2. Simplify and solve for \(x\) by multiplying both sides by \(x + 1\).

---

### e) \(\frac{2}{2x + 1} = \frac{5}{4 - x}\)

1. Cross-multiply to get rid of the fractions:
   \[
   2 \cdot (4 - x) = 5 \cdot (2x + 1)
   \]
2. Expand and solve for \(x\).

---

### f) \(\frac{5}{x - 2} = \frac{4}{x + 3}\)

1. Cross-multiply:
   \[
   5 \cdot (x + 3) = 4 \cdot (x - 2)
   \]
2. Expand and solve for \(x\).

---

Would you like detailed solutions for each part or guidance on a specific equation?

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Here are some additional questions you might find helpful:

1. How can we handle complex fractions involving multiple variables?
2. What techniques simplify solving fractional equations?
3. How does cross-multiplying work when there are fractions on both sides?
4. How can we verify solutions to ensure they don’t make any denominator zero?
5. What are common pitfalls to avoid when solving algebraic equations?

**Tip:** Always check for extraneous solutions when solving equations involving fractions.

Solve each equation algebraically:

(a) \(\frac{2}{x} + \frac{5}{3} = \frac{7}{x}\)
(b) \(\frac{10}{x + 3} + \frac{10}{3} = 6\)
(c) \(\frac{2x}{x - 3} = 1 - \frac{6}{x - 3}\)
(d) \(\frac{2}{x + 1} + \frac{1}{x + 1} = 3\)
(e) \(\frac{2}{2x + 1} = \frac{5}{4 - x}\)
(f) \(\frac{5}{x - 2} = \frac{4}{x + 3}\)

Learn to solve rational equations with fractions algebraically. This guide includes step-by-step solutions for equations involving common techniques such as cross-multiplication and clearing denominators. Suitable for Grades 9-11 students.

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