Let's go through the problem step-by-step:

Part (a): Equation of the line through the given point and parallel to the given line

The given line is $x = 6$, which is a vertical line. Vertical lines have an undefined slope because they run straight up and down.

• Parallel lines share the same slope. Since the line $x = 6$ is vertical, the line parallel to it will also be vertical and will have the same form.

So, the equation of the line through the point $(-7, -2)$ and parallel to $x = 6$ is simply the vertical line that passes through $x = -7$. The equation of this line is:

$x = -7$

Part (b): Equation of the line through the given point and perpendicular to the given line

The given line is $x = 6$, which is vertical. Perpendicular lines to vertical lines are horizontal lines. Horizontal lines have a slope of 0.

The equation of a horizontal line passing through a point is given by the form $y = c$, where $c$ is the y-coordinate of the point.

• The point given is $(-7, -2)$, so the equation of the line that is perpendicular to $x = 6$ and passes through $(-7, -2)$ is:

$y = -2$

Summary of the answers:

• (a) The equation of the line parallel to $x = 6$ through the point $(-7, -2)$ is: $x = -7$.
• (b) The equation of the line perpendicular to $x = 6$ through the point $(-7, -2)$ is: $y = -2$.

Would you like any further explanation or details on these steps? Here's a tip for working with lines:

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5 Follow-up Questions:

1. What is the general form of the equation for a vertical line?
2. How do we find the equation of a line parallel to a given line if it is not vertical?
3. How do we calculate the slope of a line that is perpendicular to a given line?
4. What does it mean for two lines to be perpendicular in terms of their slopes?
5. How can you find the equation of a line passing through a point with a known slope?

Tip:

When working with parallel lines, remember that they share the same slope. For vertical lines, their slope is undefined, so parallel vertical lines will have the same x-value. Similarly, for horizontal lines, the slope is zero, and parallel horizontal lines will have the same y-value.