To solve the problem, we need to determine which lines are perpendicular by analyzing their slopes.

Step 1: Extract the equations and rewrite them as necessary

The given equations are:

1. $y = -3$: A horizontal line with a slope of $0$.
2. $y = \frac{1}{3}x - 1$: A line with a slope of $\frac{1}{3}$.
3. $x = -7$: A vertical line with an undefined slope.
4. $y + 3 = -3(x - 5)$: Rewrite in slope-intercept form: $y + 3 = -3x + 15 \implies y = -3x + 12$ This is a line with a slope of $-3$.

Step 2: Analyze perpendicularity

Two lines are perpendicular if the product of their slopes equals $-1$, or if one is vertical ($x = c$) and the other is horizontal ($y = c$).

1. $y = -3$ (slope = $0$) and $x = -7$ (vertical line):
   • These lines are perpendicular because one is vertical and the other is horizontal.
2. $y = \frac{1}{3}x - 1$ (slope = $\frac{1}{3}$) and $y = -3x + 12$ (slope = $-3$):
   • The product of their slopes is: $\frac{1}{3} \times -3 = -1$ So these lines are perpendicular.

Step 3: Verify answer choices

• a. $y = \frac{1}{3}x - 1$ and $y + 3 = -3(x - 5)$ are perpendicular: Correct.
• b. None of the lines are perpendicular: Incorrect.
• c. $y = -3$ and $x = -7$ are perpendicular; $y = \frac{1}{3}x - 1$ and $y + 3 = -3(x - 5)$ are perpendicular: Correct.
• d. $y = -3$ and $x = -7$ are perpendicular: Partially correct but incomplete.

Correct answer: c.

Similar Problems

1. Problem 1: Identify the perpendicular lines from the following:

   • $y = 2x + 5$
   • $y = -\frac{1}{2}x + 3$
   • $y = 4$
   • $x = -2$

   Solution:

   • $y = 2x + 5$ and $y = -\frac{1}{2}x + 3$: The product of slopes is: $2 \times -\frac{1}{2} = -1$ So they are perpendicular.
   • $y = 4$ (slope = $0$) and $x = -2$ (vertical): Perpendicular.
   • Correct answer: $y = 2x + 5$ and $y = -\frac{1}{2}x + 3$; $y = 4$ and $x = -2$.

2. Problem 2: Which lines are perpendicular?

   • $y = -4x + 1$
   • $y = \frac{1}{4}x - 2$
   • $y = 2$
   • $x = 3$

   Solution:

   • $y = -4x + 1$ and $y = \frac{1}{4}x - 2$: The product of slopes is: $-4 \times \frac{1}{4} = -1$ So they are perpendicular.
   • $y = 2$ (slope = $0$) and $x = 3$ (vertical): Perpendicular.
   • Correct answer: $y = -4x + 1$ and $y = \frac{1}{4}x - 2$; $y = 2$ and $x = 3$.

Let me know if you have any questions or need more examples! Here’s one tip: Always rewrite equations into slope-intercept form ($y = mx + b$) for easy identification of slopes.