The image asks to "Tentukan Hubungan tiap pasang garis" (Determine the relationship of each pair of lines). The pairs of lines are given for three cases:

1. a) g1: $4y = 8x + 12$ and g2: $y = -2x + 1$
2. b) g1: $y = -2x + 8$ and g2: $-2y - 4x + 1 = 0$
3. c) g1: $y = 3x - 1$ and g2: $y = -\frac{1}{3}x + 9$

To solve this, we need to determine whether the lines are:

• Parallel: If the slopes are the same.
• Perpendicular: If the slopes are negative reciprocals of each other.
• Intersecting: If the slopes are different and not perpendicular.

Let's break down each case.

a) $4y = 8x + 12$ and $y = -2x + 1$

• g1: $4y = 8x + 12$
  Divide by 4:
  $y = 2x + 3$

• g2: Already given as $y = -2x + 1$

• Slopes:
  Slope of g1 is $2$, and slope of g2 is $-2$.
  Since the slopes are not equal, the lines are not parallel.
  Since $2 \times (-2) = -4$, they are not perpendicular either.
  Therefore, the lines are intersecting.

b) $y = -2x + 8$ and $-2y - 4x + 1 = 0$

• g1: Already given as $y = -2x + 8$

• g2: $-2y - 4x + 1 = 0$
  Rearrange this equation to the slope-intercept form $y = mx + c$:
  $-2y = 4x - 1$ Divide by -2: $y = -2x + \frac{1}{2}$

• Slopes:
  Slope of g1 is $-2$, and slope of g2 is also $-2$.
  Since the slopes are equal, the lines are parallel.

c) $y = 3x - 1$ and $y = -\frac{1}{3}x + 9$

• g1: Already given as $y = 3x - 1$

• g2: Already given as $y = -\frac{1}{3}x + 9$

• Slopes:
  Slope of g1 is $3$, and slope of g2 is $-\frac{1}{3}$.
  Since $3 \times -\frac{1}{3} = -1$, the lines are perpendicular.

Summary:

• a) The lines are intersecting.
• b) The lines are parallel.
• c) The lines are perpendicular.

Would you like further details on any part of this explanation?

Here are 5 related questions to expand on this topic:

1. How can we find the point of intersection of two intersecting lines?
2. What conditions must two lines satisfy to be coincident?
3. How do we determine the angle between two lines in the coordinate plane?
4. Can parallel lines ever intersect in non-Euclidean geometry?
5. How can we graph two lines to visually verify their relationship?

Tip: When solving for the relationship between two lines, always start by putting them in slope-intercept form $y = mx + c$ to easily compare slopes.