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Let’s go step-by-step to solve the question you uploaded.

The task asks for equations of lines that are:
- **Parallel** to the given lines.
- **Passing through specific points**.

Recall that **parallel lines** have the **same slope**. Let’s analyze each case:

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**a.** \( y = \frac{1}{2}x + 4 \); through \( (3, 5) \)  
- The slope is \( \frac{1}{2} \).  
- Using the **point-slope form**:  
  \( y - y_1 = m (x - x_1) \), where \( (x_1, y_1) = (3, 5) \) and \( m = \frac{1}{2} \).  
  Substituting:  
  \( y - 5 = \frac{1}{2} (x - 3) \)  
  \( y - 5 = \frac{1}{2}x - \frac{3}{2} \)  
  \( y = \frac{1}{2}x - \frac{3}{2} + 5 \)  
  \( y = \frac{1}{2}x + \frac{7}{2} \)  

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**b.** \( y = -4x + 3 \); through \( (-2, -1) \)  
- The slope is \( -4 \).  
- Using point-slope form:  
  \( y - (-1) = -4 (x - (-2)) \)  
  \( y + 1 = -4 (x + 2) \)  
  \( y + 1 = -4x - 8 \)  
  \( y = -4x - 9 \)  

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**c.** \( x + y - 2 = 0 \); through \( (-5, 0) \)  
- Rewrite the equation in slope-intercept form:  
  \( y = -x + 2 \).  
  So, the slope is \( -1 \).  
- Using point-slope form:  
  \( y - 0 = -1 (x - (-5)) \)  
  \( y = -1 (x + 5) \)  
  \( y = -x - 5 \)  

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**d.** \( y - 3x = 1 \); through \( (4, -1) \)  
- Rewrite in slope-intercept form:  
  \( y = 3x + 1 \).  
  So, the slope is \( 3 \).  
- Using point-slope form:  
  \( y - (-1) = 3 (x - 4) \)  
  \( y + 1 = 3x - 12 \)  
  \( y = 3x - 13 \)  

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### Final Results:
- **a.** \( y = \frac{1}{2}x + \frac{7}{2} \)  
- **b.** \( y = -4x - 9 \)  
- **c.** \( y = -x - 5 \)  
- **d.** \( y = 3x - 13 \)  

Let me know if you have any questions or need further steps explained!

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### Related Questions:
1. How do we find the slope of a line from a given equation?
2. What is the difference between slope-intercept form and point-slope form?
3. Can two lines be parallel if they have different y-intercepts?
4. How can we verify if two lines are parallel given their equations?
5. What is the general form of a linear equation in two variables?

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**Tip:** Parallel lines always have the **same slope** but may differ in their y-intercepts.

Tentukan persamaan garis yang sejajar dengan garis-garis berikut dan melalui titik yang ditentukan berikutnya!

Pelajari bagaimana menentukan persamaan garis yang sejajar dengan garis yang sudah diketahui dan melalui titik tertentu. Soal ini melibatkan konsep garis sejajar dan persamaan garis dalam bentuk point-slope dan slope-intercept. Cocok untuk siswa kelas 9-10.

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