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!