Question

the equation of line $r$ is $y - 9 = 4(x - 4)$. perpendicular to line $r$ is line $s$, which passes through the point $(6, -5)$. what is the equation of line $s$?
write the equation in slope-intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line $r$

The line $r$ is in point-slope form $y-y_1=m(x-x_1)$, so its slope $m_r=4$.

Step2: Find slope of line $s$

Perpendicular slopes are negative reciprocals: $m_s=-\frac{1}{m_r}=-\frac{1}{4}$

Step3: Use point-slope for line $s$

Substitute $m_s=-\frac{1}{4}$ and $(6,-5)$ into $y-y_1=m(x-x_1)$:
$y - (-5) = -\frac{1}{4}(x - 6)$

Step4: Simplify to slope-intercept form

Simplify and solve for $y$:
$y + 5 = -\frac{1}{4}x + \frac{6}{4}$
$y = -\frac{1}{4}x + \frac{3}{2} - 5$
$y = -\frac{1}{4}x + \frac{3}{2} - \frac{10}{2}$
$y = -\frac{1}{4}x - \frac{7}{2}$

Answer:

$y = -\frac{1}{4}x - \frac{7}{2}$