Question

the equation for line k can be written as $y - 8 = -\frac{8}{3}(x + 6)$. line $\ell$ includes the point $(-8, -5)$ and is perpendicular to line k. what is the equation of line $\ell$?
write the equation in slope-intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Identify slope of line k

The given line k is in point-slope form $y - y_1 = m(x - x_1)$, so its slope $m_k = -\frac{8}{3}$.

Step2: Find slope of line ℓ

Perpendicular slopes are negative reciprocals.
$m_\ell = \frac{3}{8}$

Step3: Use point-slope for line ℓ

Substitute $m_\ell = \frac{3}{8}$ and point $(-8, -5)$ into $y - y_1 = m(x - x_1)$.
$y - (-5) = \frac{3}{8}(x - (-8))$

Step4: Simplify to slope-intercept form

Simplify the equation and solve for y.
$y + 5 = \frac{3}{8}(x + 8)$
$y + 5 = \frac{3}{8}x + 3$
$y = \frac{3}{8}x + 3 - 5$
$y = \frac{3}{8}x - 2$

Answer:

$y = \frac{3}{8}x - 2$