https://file.aiquickdraw.com/mathbot/file/17261070573067b8wmhys.jpg

Let's solve each equation one by one.

---

### **29. \( 8x - 4 = 3x + 1 \)**

1. Subtract \( 3x \) from both sides:
   \[
   8x - 3x - 4 = 1 \quad \Rightarrow \quad 5x - 4 = 1
   \]
2. Add 4 to both sides:
   \[
   5x = 5
   \]
3. Divide by 5:
   \[
   x = 1
   \]

---

### **30. \( -2(5d - 8) = 20 \)**

1. Distribute the \( -2 \):
   \[
   -10d + 16 = 20
   \]
2. Subtract 16 from both sides:
   \[
   -10d = 4
   \]
3. Divide by \( -10 \):
   \[
   d = -\frac{2}{5}
   \]

---

### **31. \( 7r + 21 = 49r \)**

1. Subtract \( 7r \) from both sides:
   \[
   21 = 42r
   \]
2. Divide by 42:
   \[
   r = \frac{1}{2}
   \]

---

### **32. \( -9g - 3 = -3(3g + 2) \)**

1. Distribute the \( -3 \):
   \[
   -9g - 3 = -9g - 6
   \]
2. Add \( 9g \) to both sides:
   \[
   -3 = -6
   \]
This is a false statement, which means **no solution**.

---

### **33. \( 5(3x - 2) = 5(4x + 1) \)**

1. Distribute both sides:
   \[
   15x - 10 = 20x + 5
   \]
2. Subtract \( 15x \) from both sides:
   \[
   -10 = 5x + 5
   \]
3. Subtract 5 from both sides:
   \[
   -15 = 5x
   \]
4. Divide by 5:
   \[
   x = -3
   \]

---

### **34. \( 3d - 4 + d = 8d - (-12) \)**

1. Simplify both sides:
   \[
   4d - 4 = 8d + 12
   \]
2. Subtract \( 4d \) from both sides:
   \[
   -4 = 4d + 12
   \]
3. Subtract 12 from both sides:
   \[
   -16 = 4d
   \]
4. Divide by 4:
   \[
   d = -4
   \]

---

### **35. \( f - 6 = -2f + 3(f - 2) \)**

1. Distribute on the right-hand side:
   \[
   f - 6 = -2f + 3f - 6
   \]
2. Simplify:
   \[
   f - 6 = f - 6
   \]
This is true for all \( f \), so the equation has **infinitely many solutions**.

---

### **36. \( -2(y - 1) = 4y - (y + 2) \)**

1. Distribute both sides:
   \[
   -2y + 2 = 4y - y - 2
   \]
2. Simplify the right-hand side:
   \[
   -2y + 2 = 3y - 2
   \]
3. Add \( 2y \) to both sides:
   \[
   2 = 5y - 2
   \]
4. Add 2 to both sides:
   \[
   4 = 5y
   \]
5. Divide by 5:
   \[
   y = \frac{4}{5}
   \]

---

Let me know if you'd like further clarification on any of these! Here are 5 related questions to extend understanding:

1. How can you verify your solution when solving linear equations?
2. What does it mean when an equation has infinitely many solutions?
3. How do you solve linear equations with fractions?
4. Can a linear equation have no solutions? Why?
5. What is the importance of simplifying equations before solving?

**Tip:** Always check your solution by plugging it back into the original equation to confirm it works.

Solve the following algebraic equations: 29. 8x - 4 = 3x + 1, 30. -2(5d - 8) = 20, 31. 7r + 21 = 49r, 32. -9g - 3 = -3(3g + 2), 33. 5(3x - 2) = 5(4x + 1), 34. 3d - 4 + d = 8d - (-12), 35. f - 6 = -2f + 3(f - 2), 36. -2(y - 1) = 4y - (y + 2)

Learn how to solve multi-step linear equations, including distributive property and algebraic manipulation. Problems include equations with no solution, infinite solutions, and standard variable isolation techniques.

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