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The image shows two related geometry problems about lines and their slopes.

1. **Line A** is given by the equation \( y = \frac{1}{2}x - 2 \). 
   - **Line B** is parallel to Line A and passes through the point \( (6, -1) \).
     - We are tasked with finding the slope \( m_B \) for Line B and writing its equation in both point-slope and slope-intercept forms.
  
2. **Line C** is perpendicular to Line A and passes through the point \( (3, 4) \).
   - We need to determine the slope \( m_C \) and express the equation for Line C in point-slope and slope-intercept forms.

### Solution:

1. **Line B (Parallel to Line A)**
   - Since Line B is parallel to Line A, the slope of Line B is the same as the slope of Line A.
     \[
     m_B = \frac{1}{2}
     \]
   - **Point-slope form for Line B** using \( (x_1, y_1) = (6, -1) \):
     \[
     y - (-1) = \frac{1}{2}(x - 6)
     \]
     Simplifying:
     \[
     y + 1 = \frac{1}{2}(x - 6)
     \]
   - **Slope-intercept form for Line B**:
     Start from point-slope form:
     \[
     y + 1 = \frac{1}{2}(x - 6)
     \]
     \[
     y + 1 = \frac{1}{2}x - 3
     \]
     Subtract 1 from both sides:
     \[
     y = \frac{1}{2}x - 4
     \]
  
2. **Line C (Perpendicular to Line A)**
   - The slope of Line C, \( m_C \), is the negative reciprocal of the slope of Line A. Since the slope of Line A is \( \frac{1}{2} \), the slope of Line C is:
     \[
     m_C = -2
     \]
   - **Point-slope form for Line C** using \( (3, 4) \):
     \[
     y - 4 = -2(x - 3)
     \]
   - **Slope-intercept form for Line C**:
     Expand the point-slope form:
     \[
     y - 4 = -2(x - 3)
     \]
     \[
     y - 4 = -2x + 6
     \]
     Add 4 to both sides:
     \[
     y = -2x + 10
     \]

Let me know if you need further details or clarifications.

### 5 Related Questions:
1. How can you determine if two lines are perpendicular or parallel just by looking at their slopes?
2. What are the properties of point-slope form, and when is it most useful?
3. Can you find the equation of a line given two points on that line?
4. How would the equation change if Line B passed through a different point?
5. What is the significance of the slope-intercept form in understanding a line's behavior?

### Tip:
Remember that parallel lines always have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other!

Line A is given by the equation y = (1/2)x - 2. Line B is parallel to Line A and passes through (6, -1). Line C is perpendicular to Line A and passes through (3, 4). Find the slopes of Line B and Line C, and express their equations in point-slope and slope-intercept forms.

This problem involves finding the slope of Line B, which is parallel to Line A (y = (1/2)x - 2), and Line C, which is perpendicular to Line A. The equations of the lines are written in both point-slope and slope-intercept forms. Suitable for students in Grades 9-11.

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