https://file.aiquickdraw.com/mathbot/file/1726160496470hmjm4czb.jpg

Let's solve the equations step by step from the image:

### 1) \( 3x - 12 = 18 \)

- Add 12 to both sides: 
  \[ 3x = 18 + 12 \]
  \[ 3x = 30 \]
- Divide by 3:
  \[ x = \frac{30}{3} = 10 \]

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

- Multiply both sides by 4:
  \[ x = -5 \times 4 \]
  \[ x = -20 \]

### 3) \( \frac{5}{x} = 10 \)

- Multiply both sides by \(x\):
  \[ 5 = 10x \]
- Divide by 10:
  \[ x = \frac{5}{10} = 0.5 \]

### 4) \( 2(x + 6) = 14 \)

- Divide by 2 on both sides:
  \[ x + 6 = \frac{14}{2} = 7 \]
- Subtract 6:
  \[ x = 7 - 6 = 1 \]

### 5) \( \frac{x}{4} + 5 = 20 \)

- Subtract 5 from both sides:
  \[ \frac{x}{4} = 20 - 5 \]
  \[ \frac{x}{4} = 15 \]
- Multiply by 4:
  \[ x = 15 \times 4 = 60 \]

### 6) \( 4x - 16 = 40 \)

- Add 16 to both sides:
  \[ 4x = 40 + 16 \]
  \[ 4x = 56 \]
- Divide by 4:
  \[ x = \frac{56}{4} = 14 \]

### 7) \( \frac{3}{4}x + \frac{5}{6}x = 10 \)

- Find a common denominator for \( \frac{3}{4} \) and \( \frac{5}{6} \). The least common denominator (LCD) is 12:
  \[ \frac{3}{4}x = \frac{9}{12}x \quad \text{and} \quad \frac{5}{6}x = \frac{10}{12}x \]
- Combine the fractions:
  \[ \frac{9}{12}x + \frac{10}{12}x = \frac{19}{12}x \]
  \[ \frac{19}{12}x = 10 \]
- Multiply both sides by 12:
  \[ 19x = 120 \]
- Divide by 19:
  \[ x = \frac{120}{19} \approx 6.32 \]

### 8) \( 2.9x - 1.9x + 15 = 0 \)

- Combine like terms:
  \[ (2.9 - 1.9)x + 15 = 0 \]
  \[ 1x + 15 = 0 \]
- Subtract 15:
  \[ x = -15 \]

Would you like further clarifications on any of these problems?

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Here are 5 related questions to deepen your understanding:

1. How do you solve linear equations with fractions?
2. What is the method for solving equations involving variables on both sides?
3. How do you handle equations with decimals?
4. What is the process for solving equations with parentheses?
5. How do you find the least common denominator when working with fractional coefficients?

**Tip:** When working with fractional equations, it can help to clear fractions by multiplying both sides of the equation by the least common denominator.

Solve the system of linear equations shown in the image, including fractions and variables.

Learn how to solve linear equations step-by-step, including handling fractions and decimal terms. This problem demonstrates key algebraic methods, such as simplification, combining like terms, and solving for x. Suitable for Grades 6-8.

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