Sure! Let's go through the steps to find the equation of a line that passes through the point [tex]\((1, -5)\)[/tex] and is perpendicular to the line [tex]\(y = \frac{1}{8}x + 2\)[/tex].
1. Identify the slope of the given line:
- The slope of the given line [tex]\(y = \frac{1}{8}x + 2\)[/tex] is [tex]\(\frac{1}{8}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
- Therefore, the slope of the perpendicular line is [tex]\(-\frac{1}{(\frac{1}{8})} = -8\)[/tex].
3. Use the point-slope form of the line equation:
- The point-slope form of the equation of a line is given by [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope of the line.
- In this case, the point [tex]\((1, -5)\)[/tex] is on the line, and the slope [tex]\(m = -8\)[/tex].
4. Substitute the point [tex]\((1, -5)\)[/tex] and slope [tex]\(-8\)[/tex] into the point-slope form equation:
[tex]\[
y - (-5) = -8(x - 1)
\][/tex]
[tex]\[
y + 5 = -8(x - 1)
\][/tex]
5. Simplify the equation:
[tex]\[
y + 5 = -8x + 8
\][/tex]
[tex]\[
y = -8x + 8 - 5
\][/tex]
[tex]\[
y = -8x + 3
\][/tex]
So, the equation of the line that passes through the point [tex]\((1, -5)\)[/tex] and is perpendicular to the line [tex]\(y = \frac{1}{8}x + 2\)[/tex] is:
[tex]\[ y = -8x + 3 \][/tex]
