Let's start by addressing each question from the image you provided.

22. Determining if the two lines are parallel, perpendicular, or neither:

We are given two equations:

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

To determine if the lines are parallel, perpendicular, or neither, we need to compare the slopes of the two lines.

• The first equation is already in slope-intercept form, $y = mx + b$, where $m$ is the slope.

  • For $y = \frac{1}{2}x - 3$, the slope is $\frac{1}{2}$.

• The second equation is also in slope-intercept form:

  • For $y = -2x + 4$, the slope is $-2$.

Now, let's compare the slopes:

• If the slopes are equal, the lines are parallel.
• If the product of the slopes is $-1$, the lines are perpendicular.
• If neither of the above conditions hold, the lines are neither parallel nor perpendicular.

Let's check:

• The slopes are $\frac{1}{2}$ and $-2$.
• The product of these slopes is $\frac{1}{2} \times (-2) = -1$.

Since the product is $-1$, the two lines are perpendicular.

23. Equation of the horizontal line through the point (1, 5):

A horizontal line has a slope of 0. The equation of a horizontal line is of the form $y = b$, where $b$ is the y-coordinate of any point on the line.

• Since the point (1, 5) is on the line, the equation of the horizontal line is: $y = 5$

24. Equation of the vertical line through the point (1, 5):

A vertical line has an undefined slope. The equation of a vertical line is of the form $x = a$, where $a$ is the x-coordinate of any point on the line.

• Since the point (1, 5) is on the line, the equation of the vertical line is: $x = 1$

Would you like further details or clarification on any of these steps? Here are a few related questions:

1. How do we find the slope of a line from its equation?
2. What is the geometric interpretation of two lines being perpendicular?
3. How can we determine if two lines are neither parallel nor perpendicular?
4. Can a line be both horizontal and vertical at the same time?
5. How would the equations change if the point given were different?

Tip: When comparing slopes of two lines, remember that lines are parallel if their slopes are identical and perpendicular if the product of their slopes equals $-1$.