Sure! Let's find the equation of the line that passes through the point [tex]\((-6, -2)\)[/tex] and is perpendicular to the line [tex]\(y = 4\)[/tex].
1. Identify the slope of the given line:
- The equation [tex]\(y = 4\)[/tex] represents a horizontal line because it does not depend on [tex]\(x\)[/tex]. The slope of a horizontal line is [tex]\(0\)[/tex].
2. Determine the slope of the perpendicular line:
- A line perpendicular to a horizontal line is a vertical line. The slope of a vertical line is undefined.
3. Find the equation of the perpendicular line:
- Vertical lines have equations of the form [tex]\(x = c\)[/tex], where [tex]\(c\)[/tex] is a constant representing the x-coordinate of every point on the line. Since the line must pass through [tex]\((-6, -2)\)[/tex], the x-coordinate of that point will be our constant [tex]\(c\)[/tex].
4. Write the equation:
- Therefore, the equation of the vertical line passing through the point [tex]\((-6, -2)\)[/tex] is [tex]\(x = -6\)[/tex].
In conclusion, the equation of the line passing through [tex]\((-6, -2)\)[/tex] and perpendicular to [tex]\(y = 4\)[/tex] is:
[tex]\[ \boxed{x = -6} \][/tex]
