Let's analyze each question one by one.

Question 21: Parallel Line

To find a line that is parallel, we need to determine the slope of the given line and then identify which option has the same slope.

1. Identify the line in the graph, calculate its slope.

   • From the graph, the line passes through two points (for example, (-3, 3) and (3, -3)).
   • Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{3 - (-3)} = \frac{-6}{6} = -1$.

2. Rewrite each option in slope-intercept form $y = mx + b$:

   • Option A: $4x + 5y = -10 \Rightarrow y = -\frac{4}{5}x - 2$ (slope = $-\frac{4}{5}$)
   • Option B: $4x - 5y = 0 \Rightarrow y = \frac{4}{5}x$ (slope = $\frac{4}{5}$)
   • Option C: $5x + 4y = 24 \Rightarrow y = -\frac{5}{4}x + 6$ (slope = $-\frac{5}{4}$)
   • Option D: $5x - 4y = -8 \Rightarrow y = \frac{5}{4}x + 2$ (slope = $\frac{5}{4}$)

Since none of these slopes match $-1$, it appears no correct option is given in this case for parallel.

Question 22: Perpendicular Line

For perpendicularity, we need the negative reciprocal of the original line's slope.

1. Given slope of the line in the graph is $-1$.

   • The perpendicular slope will be $1$.

2. Check each option for a slope of $1$:

   • Option A: $x + 2y = -14 \Rightarrow y = -\frac{1}{2}x - 7$ (slope = $-\frac{1}{2}$)
   • Option B: $x - 2y = 10 \Rightarrow y = \frac{1}{2}x - 5$ (slope = $\frac{1}{2}$)
   • Option C: $2x + y = -1 \Rightarrow y = -2x - 1$ (slope = $-2$)
   • Option D: $2x - y = 6 \Rightarrow y = 2x - 6$ (slope = $2$)

No option matches the perpendicular slope either. Thus, both questions seem to lack correct answer choices based on slope requirements.

Would you like further details on these calculations?

Additional Questions:

1. How do you determine slopes from equations directly?
2. Why is the slope important in determining parallel and perpendicular lines?
3. What would be the slope for lines that are neither parallel nor perpendicular?
4. How can you verify parallelism if you know two points on each line?
5. What are typical strategies for quickly converting equations to slope-intercept form?

Tip:

For quick checks on parallel or perpendicular lines, directly compare slopes after converting to slope-intercept form.