write the equation of the line perpendicular to (y = \frac{1}{8}x + 2) through the point ((1, - 5)).

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line $y = \frac{1}{8}x+2$ is $m_1=\frac{1}{8}$. For two - perpendicular lines with slopes $m_1$ and $m_2$, $m_1\times m_2=- 1$. So, $\frac{1}{8}\times m_2=-1$, then $m_2=-8$.

Step2: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,-5)$ and $m=-8$. Substitute these values into the formula: $y-(-5)=-8(x - 1)$.

Step3: Simplify the equation

$y + 5=-8x + 8$. Then, $y=-8x+8 - 5$, so $y=-8x + 3$.

Answer:

$y=-8x + 3$