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answer questions 37, 39, 40, 42, 46, 50, and 55

Let's go through the requested problems one by one from the image you provided.

### 37. Solve the equation for \( x \):
\[
4x + 3 = 9
\]
**Solution**:
- Subtract 3 from both sides:  
  \[
  4x = 6
  \]
- Divide by 4:  
  \[
  x = \frac{6}{4} = \frac{3}{2}
  \]

### 39. Solve the equation for \( x \):
\[
\frac{3}{5} = 1 - \frac{x}{2}
\]
**Solution**:
- Subtract 1 from both sides:  
  \[
  \frac{3}{5} - 1 = -\frac{x}{2}
  \]
- Convert \( 1 \) to a fraction with denominator 5:  
  \[
  \frac{3}{5} - \frac{5}{5} = -\frac{x}{2}
  \]
- Simplify:  
  \[
  \frac{-2}{5} = -\frac{x}{2}
  \]
- Multiply both sides by -2:  
  \[
  x = \frac{4}{5}
  \]

### 40. Solve the equation:
\[
8(x + 2) + 3(x - 4) = 6
\]
**Solution**:
- Distribute the 8 and 3:  
  \[
  8x + 16 + 3x - 12 = 6
  \]
- Combine like terms:  
  \[
  11x + 4 = 6
  \]
- Subtract 4 from both sides:  
  \[
  11x = 2
  \]
- Divide by 11:  
  \[
  x = \frac{2}{11}
  \]

### 42. Solve the equation:
\[
5x - 4 = 6x - 3
\]
**Solution**:
- Subtract 5x from both sides:  
  \[
  -4 = x - 3
  \]
- Add 3 to both sides:  
  \[
  -1 = x
  \]

### 46. Solve the equation:
\[
8x - 5(3x - 7) = 2(x + 6)
\]
**Solution**:
- Distribute the -5 and 2:  
  \[
  8x - 15x + 35 = 2x + 12
  \]
- Combine like terms:  
  \[
  -7x + 35 = 2x + 12
  \]
- Add 7x to both sides:  
  \[
  35 = 9x + 12
  \]
- Subtract 12 from both sides:  
  \[
  23 = 9x
  \]
- Divide by 9:  
  \[
  x = \frac{23}{9}
  \]

### 50. Solve the equation:
\[
4m - 2n = 3(2m + n) + 1
\]
**Solution**:
- Distribute the 3:  
  \[
  4m - 2n = 6m + 3n + 1
  \]
- Move all terms involving \( m \) and \( n \) to one side:  
  \[
  4m - 6m - 2n - 3n = 1
  \]
- Simplify:  
  \[
  -2m - 5n = 1
  \]

### 55. Solve the equation:
\[
8k - 2 = 5(k + 3)
\]
**Solution**:
- Distribute the 5:  
  \[
  8k - 2 = 5k + 15
  \]
- Subtract 5k from both sides:  
  \[
  3k - 2 = 15
  \]
- Add 2 to both sides:  
  \[
  3k = 17
  \]
- Divide by 3:  
  \[
  k = \frac{17}{3}
  \]

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Would you like any further details on these solutions or explanations?

Here are some related questions you might consider:
1. How do you solve an equation with variables on both sides?
2. What is the significance of isolating the variable in an equation?
3. How does distribution work in algebraic expressions?
4. Why do we need to combine like terms in an equation?
5. What are some common mistakes to avoid when solving linear equations?

**Tip:** Always double-check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Explore step-by-step solutions to linear equations including 4x + 3 = 9, 3/5 = 1 - x/2, and more. Learn the methods and techniques needed to solve these equations effectively. Suitable for high school students and anyone learning algebra.

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