Solving Linear Equations worksheet for Grade 4 students, focusing on simplifying fractions and decimals.

Problem: Solve the linear equations and leave your answers as simplified fractions or decimals.

#### Section A

1. Equation: $\frac{2x + 5}{3} = 11$

- Step 1: Eliminate the denominator by multiplying both sides by 3:
$$
2x + 5 = 33
$$

- Step 2: Isolate $x$ by subtracting 5 from both sides:
$$
2x = 28
$$

- Step 3: Solve for $x$ by dividing both sides by 2:
$$
x = 14
$$

- Answer: $\boxed{14}$

---

2. Equation: $\frac{8 - 3x}{2} = 5$

- Step 1: Eliminate the denominator by multiplying both sides by 2:
$$
8 - 3x = 10
$$

- Step 2: Isolate $x$ by subtracting 8 from both sides:
$$
-3x = 2
$$

- Step 3: Solve for $x$ by dividing both sides by $-3$:
$$
x = -\frac{2}{3}
$$

- Answer: $\boxed{-\frac{2}{3}}$

---

3. Equation: $\frac{5 - 9x}{6} = -2$

- Step 1: Eliminate the denominator by multiplying both sides by 6:
$$
5 - 9x = -12
$$

- Step 2: Isolate $x$ by subtracting 5 from both sides:
$$
-9x = -17
$$

- Step 3: Solve for $x$ by dividing both sides by $-9$:
$$
x = \frac{17}{9}
$$

- Answer: $\boxed{\frac{17}{9}}$

---

4. Equation: $\frac{7x + 6}{3} - 9 = -12$

- Step 1: Eliminate the constant term by adding 9 to both sides:
$$
\frac{7x + 6}{3} = -3
$$

- Step 2: Eliminate the denominator by multiplying both sides by 3:
$$
7x + 6 = -9
$$

- Step 3: Isolate $x$ by subtracting 6 from both sides:
$$
7x = -15
$$

- Step 4: Solve for $x$ by dividing both sides by 7:
$$
x = -\frac{15}{7}
$$

- Answer: $\boxed{-\frac{15}{7}}$

---

5. Equation: $8x + \frac{1 - 4x}{8} = 7$

- Step 1: Eliminate the fraction by multiplying every term by 8:
$$
8 \cdot 8x + (1 - 4x) = 8 \cdot 7
$$
$$
64x + 1 - 4x = 56
$$

- Step 2: Combine like terms:
$$
60x + 1 = 56
$$

- Step 3: Isolate $x$ by subtracting 1 from both sides:
$$
60x = 55
$$

- Step 4: Solve for $x$ by dividing both sides by 60:
$$
x = \frac{55}{60} = \frac{11}{12}
$$

- Answer: $\boxed{\frac{11}{12}}$

---

6. Equation: $\frac{5}{x} = -6$

- Step 1: Eliminate the denominator by multiplying both sides by $x$:
$$
5 = -6x
$$

- Step 2: Solve for $x$ by dividing both sides by $-6$:
$$
x = -\frac{5}{6}
$$

- Answer: $\boxed{-\frac{5}{6}}$

---

7. Equation: $\frac{11}{4x} + 9 = 3$

- Step 1: Isolate the fraction by subtracting 9 from both sides:
$$
\frac{11}{4x} = -6
$$

- Step 2: Eliminate the denominator by multiplying both sides by $4x$:
$$
11 = -6 \cdot 4x
$$
$$
11 = -24x
$$

- Step 3: Solve for $x$ by dividing both sides by $-24$:
$$
x = -\frac{11}{24}
$$

- Answer: $\boxed{-\frac{11}{24}}$

---

8. Equation: $5 - \frac{3x}{4} = 8x$

- Step 1: Eliminate the fraction by multiplying every term by 4:
$$
4 \cdot 5 - 3x = 4 \cdot 8x
$$
$$
20 - 3x = 32x
$$

- Step 2: Combine like terms by adding $3x$ to both sides:
$$
20 = 35x
$$

- Step 3: Solve for $x$ by dividing both sides by 35:
$$
x = \frac{20}{35} = \frac{4}{7}
$$

- Answer: $\boxed{\frac{4}{7}}$

---

9. Equation: $2 + \frac{4x}{3} - 7 = 1$

- Step 1: Simplify the left-hand side:
$$
\frac{4x}{3} - 5 = 1
$$

- Step 2: Isolate the fraction by adding 5 to both sides:
$$
\frac{4x}{3} = 6
$$

- Step 3: Eliminate the denominator by multiplying both sides by 3:
$$
4x = 18
$$

- Step 4: Solve for $x$ by dividing both sides by 4:
$$
x = \frac{18}{4} = \frac{9}{2}
$$

- Answer: $\boxed{\frac{9}{2}}$

---

10. Equation: $4 - \frac{3x}{2} = 3x + 5$

- Step 1: Eliminate the fraction by multiplying every term by 2:
$$
2 \cdot 4 - 3x = 2 \cdot 3x + 2 \cdot 5
$$
$$
8 - 3x = 6x + 10
$$

- Step 2: Combine like terms by adding $3x$ to both sides:
$$
8 = 9x + 10
$$

- Step 3: Isolate $x$ by subtracting 10 from both sides:
$$
-2 = 9x
$$

- Step 4: Solve for $x$ by dividing both sides by 9:
$$
x = -\frac{2}{9}
$$

- Answer: $\boxed{-\frac{2}{9}}$

---

11. Equation: $6 - \frac{2}{x} = 10$

- Step 1: Isolate the fraction by subtracting 6 from both sides:
$$
-\frac{2}{x} = 4
$$

- Step 2: Eliminate the negative sign by multiplying both sides by $-1$:
$$
\frac{2}{x} = -4
$$

- Step 3: Eliminate the denominator by multiplying both sides by $x$:
$$
2 = -4x
$$

- Step 4: Solve for $x$ by dividing both sides by $-4$:
$$
x = -\frac{2}{4} = -\frac{1}{2}
$$

- Answer: $\boxed{-\frac{1}{2}}$

---

12. Equation: $4 - \frac{2x}{9} + x = -1$

- Step 1: Combine like terms on the left-hand side:
$$
4 + \left(x - \frac{2x}{9}\right) = -1
$$

- Step 2: Rewrite $x$ as $\frac{9x}{9}$ to combine the terms:
$$
4 + \left(\frac{9x}{9} - \frac{2x}{9}\right) = -1
$$
$$
4 + \frac{7x}{9} = -1
$$

- Step 3: Isolate the fraction by subtracting 4 from both sides:
$$
\frac{7x}{9} = -5
$$

- Step 4: Eliminate the denominator by multiplying both sides by 9:
$$
7x = -45
$$

- Step 5: Solve for $x$ by dividing both sides by 7:
$$
x = -\frac{45}{7}
$$

- Answer: $\boxed{-\frac{45}{7}}$

---

Section B

1. Equation: $4(2x - 3) = 8(2x + 5)$

- Step 1: Distribute the constants on both sides:
$$
8x - 12 = 16x + 40
$$

- Step 2: Combine like terms by subtracting $8x$ from both sides:
$$
-12 = 8x + 40
$$

- Step 3: Isolate $x$ by subtracting 40 from both sides:
$$
-52 = 8x
$$

- Step 4: Solve for $x$ by dividing both sides by 8:
$$
x = -\frac{52}{8} = -\frac{13}{2}
$$

- Answer: $\boxed{-\frac{13}{2}}$

---

2. Equation: $6(4x - 5) = 5(3x - 5)$

- Step 1: Distribute the constants on both sides:
$$
24x - 30 = 15x - 25
$$

- Step 2: Combine like terms by subtracting $15x$ from both sides:
$$
9x - 30 = -25
$$

- Step 3: Isolate $x$ by adding 30 to both sides:
$$
9x = 5
$$

- Step 4: Solve for $x$ by dividing both sides by 9:
$$
x = \frac{5}{9}
$$

- Answer: $\boxed{\frac{5}{9}}$

---

3. Equation: $7(4 - 3x) = 2(8x - 9) + 6$

- Step 1: Distribute the constants on both sides:
$$
28 - 21x = 16x - 18 + 6
$$

- Step 2: Simplify the right-hand side:
$$
28 - 21x = 16x - 12
$$

- Step 3: Combine like terms by adding $21x$ to both sides:
$$
28 = 37x - 12
$$

- Step 4: Isolate $x$ by adding 12 to both sides:
$$
40 = 37x
$$

- Step 5: Solve for $x$ by dividing both sides by 37:
$$
x = \frac{40}{37}
$$

- Answer: $\boxed{\frac{40}{37}}$

---

4. Equation: $3(2x - 4) = 2x + 3(x + 4)$

- Step 1: Distribute the constants on both sides:
$$
6x - 12 = 2x + 3x + 12
$$

- Step 2: Simplify the right-hand side:
$$
6x - 12 = 5x + 12
$$

- Step 3: Combine like terms by subtracting $5x$ from both sides:
$$
x - 12 = 12
$$

- Step 4: Isolate $x$ by adding 12 to both sides:
$$
x = 24
$$

- Answer: $\boxed{24}$

---

Final Answers:

- Section A:
1. $\boxed{14}$
2. $\boxed{-\frac{2}{3}}$
3. $\boxed{\frac{17}{9}}$
4. $\boxed{-\frac{15}{7}}$
5. $\boxed{\frac{11}{12}}$
6. $\boxed{-\frac{5}{6}}$
7. $\boxed{-\frac{11}{24}}$
8. $\boxed{\frac{4}{7}}$
9. $\boxed{\frac{9}{2}}$
10. $\boxed{-\frac{2}{9}}$
11. $\boxed{-\frac{1}{2}}$
12. $\boxed{-\frac{45}{7}}$

- Section B:
1. $\boxed{-\frac{13}{2}}$
2. $\boxed{\frac{5}{9}}$
3. $\boxed{\frac{40}{37}}$
4. $\boxed{24}$

$\boxed{\text{All solutions are provided above.}}$