Let’s tackle each question individually.

Question 5

We need to find the equation of a line that is parallel to $y = -\frac{1}{3}x + 4$ and passes through the point $(6, 5)$.

Solution:

1. Identify the slope of the given line $y = -\frac{1}{3}x + 4$, which is $m = -\frac{1}{3}$.
2. Since parallel lines have the same slope, our new line will also have a slope of $-\frac{1}{3}$.
3. Using the point-slope form of a line equation $y - y_1 = m(x - x_1)$ with point $(6, 5)$ and slope $-\frac{1}{3}$: $y - 5 = -\frac{1}{3}(x - 6)$
4. Expand and simplify: $y - 5 = -\frac{1}{3}x + 2$ $y = -\frac{1}{3}x + 7$ So, the correct answer for Question 5 is: $\boxed{y = -\frac{1}{3}x + 7}$

Question 6

We are asked about the relationship between the graphs of the lines $y = -6$ and $x = \frac{1}{6}$.

Solution:

1. Analyze each line:
   • The line $y = -6$ is a horizontal line with a slope of $0$ (since the slope of a horizontal line is always zero).
   • The line $x = \frac{1}{6}$ is a vertical line, which has an undefined slope (as vertical lines do not have a defined slope).
2. Perpendicularity: A horizontal line and a vertical line are always perpendicular to each other.

Therefore, the correct answer for Question 6 is: $\text{The lines are perpendicular to each other because the graph of } y = -6 \text{ is a horizontal line with a slope of 0, and the graph of } x = \frac{1}{6} \text{ is a vertical line with a slope that is undefined.}$

Let me know if you'd like further clarification on any of these explanations!

Related Questions:

1. How would you find the equation of a line perpendicular to $y = -\frac{1}{3}x + 4$?
2. What is the slope of a line perpendicular to a horizontal line?
3. How can we confirm if two lines are parallel by using their equations?
4. What is the difference in interpretation between a slope of 0 and an undefined slope?
5. How can the point-slope form be useful for writing line equations passing through specific points?

Tip:

When dealing with slopes, remember that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other (assuming neither slope is zero or undefined).