Question

question
find the equation of a line perpendicular to ( y + 8 = \frac{1}{3}x ) that passes through the point ( (2,1) ).
answer
( y - 1 = -3(x - 2) )
( y + 1 = \frac{1}{3}(x - 2) )
( y - 1 = \frac{1}{3}(x - 2) )
( y + 1 = -\frac{1}{3}(x - 2) )

Explanation:

Step1: Identify given line's slope

Rewrite given line to slope-intercept form:
$y + 8 = \frac{1}{3}x \implies y = \frac{1}{3}x - 8$
Slope of given line $m_1 = \frac{1}{3}$

Step2: Find perpendicular slope

Perpendicular slopes multiply to -1:
$m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{3}} = -3$

Step3: Use point-slope form

Point-slope formula: $y - y_1 = m(x - x_1)$ with $(x_1,y_1)=(2,1)$ and $m=-3$:
$y - 1 = -3(x - 2)$

Answer:

$\boldsymbol{y - 1 = -3(x - 2)}$