Let’s work through the problems on the page step-by-step. Here's how we address each section:

28) Determine if the lines are parallel, perpendicular, or neither:

1. Line 1: $y = \frac{2}{3}x - 4$
   • Slope: $m = \frac{2}{3}$
2. Line 2: $3y = 2x - 36$
   • Rewrite as $y = \frac{2}{3}x - 12$
   • Slope: $m = \frac{2}{3}$

Conclusion: The slopes are equal, so the lines are parallel.

29) Determine if the lines are parallel, perpendicular, or neither:

1. Line 1: $5x - 4y = -2$
   • Rewrite as $y = \frac{5}{4}x + \frac{1}{2}$
   • Slope: $m = \frac{5}{4}$
2. Line 2: $5y = -4x + 35$
   • Rewrite as $y = -\frac{4}{5}x + 7$
   • Slope: $m = -\frac{4}{5}$

Conclusion: The product of the slopes is $\frac{5}{4} \cdot -\frac{4}{5} = -1$, so the lines are perpendicular.

30) Determine if the lines are parallel, perpendicular, or neither:

1. Line 1: $y = \frac{2}{7}x + 4$
   • Slope: $m = \frac{2}{7}$
2. Line 2: $7y = -2x - 7$
   • Rewrite as $y = -\frac{2}{7}x - 1$
   • Slope: $m = -\frac{2}{7}$

Conclusion: The slopes are neither equal nor negative reciprocals, so the lines are neither.

31–33) Find the distance between points or lines:

• 31) Points $x = -4$ and $x = 7$

  • Distance: $|7 - (-4)| = 11$.

• 32) Points $y = 3$ and $y = -5$

  • Distance: $|3 - (-5)| = 8$.

• 33) Points $(-4, 1)$ and $(7, -2)$

  • Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $d = \sqrt{(7 - (-4))^2 + (-2 - 1)^2} = \sqrt{11^2 + (-3)^2} = \sqrt{121 + 9} = \sqrt{130}$

Distance: $\sqrt{130}$ or approximately $11.4$.

Bonus: Distance between two lines

A) $y = 2x - 4$ and $y = 2x + 9$

• Distance between parallel lines: $d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}}$ Here, $m = 2$, $c_1 = -4$, and $c_2 = 9$: $d = \frac{|9 - (-4)|}{\sqrt{1 + 2^2}} = \frac{13}{\sqrt{5}} \approx 5.81$

B) $y = \frac{3}{4}x - 5$ and $3x - 4y = -48$

• Rewrite $3x - 4y = -48$ as $y = \frac{3}{4}x + 12$.
• Distance between parallel lines: $d = \frac{|12 - (-5)|}{\sqrt{1 + (\frac{3}{4})^2}} = \frac{17}{\sqrt{\frac{25}{16}}} = \frac{17 \cdot 4}{5} = 13.6$

Would you like any additional details or further breakdowns for any part of this problem?

Questions to Explore:

1. How do you derive the formula for the distance between two parallel lines?
2. What happens when the slopes are the same for both lines?
3. How does the perpendicular condition $m_1 \cdot m_2 = -1$ arise geometrically?
4. What is the significance of the distance formula in coordinate geometry?
5. Can the distance between two skew lines in 3D space be calculated similarly?

Tip: Always verify line equations are in slope-intercept form before comparing their slopes!