Class 8 Maths worksheet on linear equations in one variable, designed for students to practice solving equations and expressing answers as proper fractions.

Let's solve each of the linear equations step by step and provide the answers in proper fractions, as instructed.

---

1. $ \frac{5x}{3} + \frac{2}{5} = 1 $

Step 1: Subtract $ \frac{2}{5} $ from both sides:

$$
\frac{5x}{3} = 1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}
$$

Step 2: Multiply both sides by 3:

$$
5x = \frac{9}{5}
$$

Step 3: Divide by 5:

$$
x = \frac{9}{25}
$$

✔ Answer: $ \boxed{\frac{9}{25}} $

---

2. $ \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 13 $

Step 1: Find LCM of denominators: LCM(2, 3, 4) = 12

Convert all terms:

$$
\frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 13
\Rightarrow \frac{13x}{12} = 13
$$

Step 2: Multiply both sides by 12:

$$
13x = 156
\Rightarrow x = \frac{156}{13} = 12
$$

✔ Answer: $ \boxed{12} $ (or $ \frac{12}{1} $)

---

3. $ \frac{x}{2} + \frac{x}{8} = \frac{1}{8} $

Step 1: LCM of 2 and 8 is 8

$$
\frac{4x}{8} + \frac{x}{8} = \frac{1}{8} \Rightarrow \frac{5x}{8} = \frac{1}{8}
$$

Step 2: Multiply both sides by 8:

$$
5x = 1 \Rightarrow x = \frac{1}{5}
$$

✔ Answer: $ \boxed{\frac{1}{5}} $

---

4. $ \frac{2x}{3} - \frac{3x}{8} = \frac{7}{12} $

Step 1: LCM of 3 and 8 is 24

$$
\frac{16x}{24} - \frac{9x}{24} = \frac{7}{12} \Rightarrow \frac{7x}{24} = \frac{7}{12}
$$

Step 2: Multiply both sides by 24:

$$
7x = 14 \Rightarrow x = 2
$$

✔ Answer: $ \boxed{2} $

---

5. $ (x+2)(x+3) + (x-3)(x-2) - 2x(x+1) = 0 $

Step 1: Expand each term:

- $ (x+2)(x+3) = x^2 + 5x + 6 $
- $ (x-3)(x-2) = x^2 - 5x + 6 $
- $ 2x(x+1) = 2x^2 + 2x $

Now plug in:

$$
(x^2 + 5x + 6) + (x^2 - 5x + 6) - (2x^2 + 2x) = 0
$$

Simplify:

$$
x^2 + 5x + 6 + x^2 - 5x + 6 - 2x^2 - 2x = 0
\Rightarrow (2x^2 - 2x^2) + (5x - 5x - 2x) + (6 + 6) = 0
\Rightarrow -2x + 12 = 0
$$

Step 2: Solve:

$$
-2x = -12 \Rightarrow x = 6
$$

✔ Answer: $ \boxed{6} $

---

6. $ \frac{x}{2} - \frac{4}{5} + \frac{x}{5} + \frac{3x}{10} = \frac{1}{5} $

Step 1: Combine like terms. Group $ x $-terms:

$$
\left(\frac{x}{2} + \frac{x}{5} + \frac{3x}{10}\right) - \frac{4}{5} = \frac{1}{5}
$$

LCM of 2, 5, 10 is 10:

$$
\frac{5x}{10} + \frac{2x}{10} + \frac{3x}{10} = \frac{10x}{10} = x
$$

So equation becomes:

$$
x - \frac{4}{5} = \frac{1}{5}
$$

Step 2: Add $ \frac{4}{5} $ to both sides:

$$
x = \frac{1}{5} + \frac{4}{5} = 1
$$

✔ Answer: $ \boxed{1} $

---

7. $ \frac{7}{x} + 35 = \frac{1}{10} $

Step 1: Subtract 35 from both sides:

$$
\frac{7}{x} = \frac{1}{10} - 35 = \frac{1}{10} - \frac{350}{10} = -\frac{349}{10}
$$

Step 2: Take reciprocal:

$$
\frac{7}{x} = -\frac{349}{10} \Rightarrow x = 7 \cdot \left(-\frac{10}{349}\right) = -\frac{70}{349}
$$

✔ Answer: $ \boxed{-\frac{70}{349}} $

---

8. $ \frac{2x - 1}{3} - \frac{6x - 2}{5} = \frac{1}{3} $

Step 1: Multiply entire equation by LCM of 3 and 5 = 15:

$$
15 \left( \frac{2x - 1}{3} - \frac{6x - 2}{5} \right) = 15 \cdot \frac{1}{3}
\Rightarrow 5(2x - 1) - 3(6x - 2) = 5
$$

Step 2: Expand:

$$
10x - 5 - 18x + 6 = 5
\Rightarrow -8x + 1 = 5
$$

Step 3: Solve:

$$
-8x = 4 \Rightarrow x = -\frac{4}{8} = -\frac{1}{2}
$$

✔ Answer: $ \boxed{-\frac{1}{2}} $

---

9. $ 13(y - 4) - 3(y - 9) - 5(y + 4) = 0 $

Step 1: Expand:

- $ 13y - 52 $
- $ -3y + 27 $
- $ -5y - 20 $

Add them:

$$
13y - 52 - 3y + 27 - 5y - 20 = (13y - 3y - 5y) + (-52 + 27 - 20) = 5y - 45
$$

Set equal to zero:

$$
5y - 45 = 0 \Rightarrow 5y = 45 \Rightarrow y = 9
$$

✔ Answer: $ \boxed{9} $

---

10. $ \frac{2}{3}(x - 5) - \frac{1}{4}(x - 2) = \frac{9}{2} $

Step 1: Expand:

$$
\frac{2}{3}x - \frac{10}{3} - \frac{1}{4}x + \frac{2}{4} = \frac{9}{2}
\Rightarrow \left(\frac{2}{3}x - \frac{1}{4}x\right) + \left(-\frac{10}{3} + \frac{1}{2}\right) = \frac{9}{2}
$$

Step 2: Simplify $ x $-terms:

LCM of 3 and 4 is 12:

$$
\frac{8}{12}x - \frac{3}{12}x = \frac{5}{12}x
$$

Constants:

$$
-\frac{10}{3} + \frac{1}{2} = -\frac{20}{6} + \frac{3}{6} = -\frac{17}{6}
$$

So:

$$
\frac{5}{12}x - \frac{17}{6} = \frac{9}{2}
$$

Step 3: Add $ \frac{17}{6} $ to both sides:

$$
\frac{5}{12}x = \frac{9}{2} + \frac{17}{6} = \frac{27}{6} + \frac{17}{6} = \frac{44}{6} = \frac{22}{3}
$$

Step 4: Multiply both sides by $ \frac{12}{5} $:

$$
x = \frac{22}{3} \cdot \frac{12}{5} = \frac{264}{15} = \frac{88}{5}
$$

✔ Answer: $ \boxed{\frac{88}{5}} $

---

✔ Final Answers:

1. $ \frac{9}{25} $
2. $ 12 $
3. $ \frac{1}{5} $
4. $ 2 $
5. $ 6 $
6. $ 1 $
7. $ -\frac{70}{349} $
8. $ -\frac{1}{2} $
9. $ 9 $
10. $ \frac{88}{5} $

Let me know if you'd like this formatted for printing or need explanations in simpler language!