To find the equation of a line that is perpendicular to the line [tex]\( y = 5 \)[/tex] and passes through the point [tex]\((-7, -5)\)[/tex], let's follow these steps:
1. Identify the type of the given line:
- The line [tex]\( y = 5 \)[/tex] is a horizontal line. It means that for every point on this line, the [tex]\( y \)[/tex]-coordinate is always 5 irrespective of the [tex]\( x \)[/tex]-coordinate.
2. Determine the slope of the perpendicular line:
- Lines that are perpendicular to horizontal lines are vertical lines because a vertical line has undefined slope, while horizontal lines have a slope of 0.
3. Write the general form of the vertical line:
- The general form of the equation of a vertical line that passes through a point is [tex]\( x = \text{constant} \)[/tex].
4. Determine the specific constant for our line:
- Since the line must pass through the point [tex]\((-7, -5)\)[/tex], the [tex]\( x \)[/tex]-coordinate of this point is crucial. Our vertical line must have the form [tex]\( x = -7 \)[/tex] since it must pass through [tex]\((-7, -5)\)[/tex], meaning that [tex]\( x \)[/tex] must always be [tex]\(-7\)[/tex] for this line.
Therefore, the equation of the line that is perpendicular to [tex]\( y = 5 \)[/tex] and passes through the point [tex]\((-7, -5)\)[/tex] is:
[tex]\[ x = -7. \][/tex]
