Question

the equation for line $r$ can be written as $y = -\frac{4}{9}x - 2$. line $s$ is perpendicular to line $r$ and passes through $(3, 5)$. 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 perpendicular slope

The slope of line $r$ is $m_r = -\frac{4}{9}$. The slope of a perpendicular line is the negative reciprocal, so $m_s = \frac{9}{4}$.

Step2: Use point-slope form

Point-slope formula: $y - y_1 = m(x - x_1)$. Substitute $m_s=\frac{9}{4}$, $x_1=3$, $y_1=5$:
$y - 5 = \frac{9}{4}(x - 3)$

Step3: Convert to slope-intercept form

Expand and solve for $y$:
$y - 5 = \frac{9}{4}x - \frac{27}{4}$
$y = \frac{9}{4}x - \frac{27}{4} + 5$
$y = \frac{9}{4}x - \frac{27}{4} + \frac{20}{4}$
$y = \frac{9}{4}x - \frac{7}{4}$

Answer:

$y = \frac{9}{4}x - \frac{7}{4}$